2023-06-16 Resolution of Nonsingularities for Arbitrary Hyperbolic Curves.pdf

Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves Shinichi Mochizuki and Shota Tsujimura June 16, 2023 Abstract In this paper, we prove that arbitrary hyperbolic curves over p-adic lo- cal ﬁelds admit resolution of nonsingularities [“RNS”]. This result may be regarded as a generalization of results concerning resolution of nonsingu- larities obtained by A. Tamagawa and E. Lepage. Moreover, by combining our RNS result with techniques from combinatorial anabelian geometry, we prove that an absolute version of the geometrically pro-Σ Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local ﬁelds, where Σ denotes a set of prime numbers of cardinality ≥ 2 that contains p, holds. This settles one of the major open questions in anabelian geometry. Furthermore, we prove — again by applying RNS and combinatorial an- abelian geometry — that the various p-adic versions of the Grothendieck- Teichmüller group that appear in the literature in fact coincide. As a corollary, we conclude that the metrized Grothendieck-Teichmüller group is commensurably terminal in the Grothendieck-Teichmüller group. This settles a longstanding open question in combinatorial anabelian geometry. Contents Introduction 2 Notations and Conventions 11 1 Local construction of Artin-Schreier extensions in the special ﬁber 15 2020 Mathematics Subject Classiﬁcation: Primary 14H30; Secondary 14H25. Keywords and phrases: anabelian geometry; resolution of nonsingularities; abso- lute Grothendieck Conjecture; combinatorial anabelian geometry; Grothendieck-Teichmüller group; étale fundamental group; tempered fundamental group; hyperbolic curve; conﬁguration space. 1 2 Resolution of nonsingularities for arbitrary hyperbolic curves over p-adic local ﬁelds 26 3 Point-theoreticity, metric-admissibility, and arithmetic cuspi- dalization 72 References 103 Introduction Let p be a prime number; Σ a nonempty subset of the set Primes of prime numbers. For a connected noetherian scheme S, we shall write Π S for the étale fundamental group of S, relative to a suitable choice of basepoint. For any ﬁeld F of characteristic 0, any ﬁeld extension F ⊆ E, and any algebraic variety [i.e., a separated, geometrically integral scheme of ﬁnite type] Z over F , we shall def write Z E = Z × F E and denote by F an algebraic closure [well-deﬁned up to def isomorphism] of F and by G F = Gal(F /F ) the absolute Galois group of F . For any ﬁeld F of characteristic 0 and any algebraic variety Z over F , we shall write (Σ) def Π Z = Π Z /Ker(Π Z F Π Σ Z F ), where Π Z F Π Σ Z F denotes the maximal pro-Σ quotient. Here, we recall that (Σ) Π Z is often referred to as the geometrically pro-Σ fundamental group of Z. We shall write Q p for the ﬁeld of p-adic numbers; C p for the p-adic completion of Q p . We shall refer to a ﬁnite extension ﬁeld of Q p as a p-adic local ﬁeld. For any hyperbolic curve Z over either the algebraic closure of a mixed character- istic complete discrete valuation ﬁeld of residue characteristic p or the p-adic completion of such an algebraic closure, we shall write Π tp Z for the Σ-tempered fundamental group of Z, relative to a suitable choice of base- point [cf. the subsection in Notations and Conventions entitled “Fundamental groups”]. If S 1 , S 2 are schemes, then we shall write Isom(S 1 , S 2 ) for the set of isomorphisms of schemes between S 1 and S 2 . If G 1 , G 2 are proﬁnite groups, then we shall write OutIsom(G 1 , G 2 ) for the set of isomorphisms of proﬁnite groups, considered up to composition with an inner automorphism arising from an element ∈ G 2 . In the present paper, we give a complete aﬃrmative answer to the following question: 2 Does an arbitrary hyperbolic curve over a p-adic local ﬁeld admit resolution of nonsingularities? Before continuing, we review the notion of resolution of nonsingularities. Let K (⊆ K) be a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; v a valuation on a ﬁeld F that contains K. Write O K for the ring of integers of K; O v for the ring of integers determined by v; m v ⊆ O v for the maximal ideal of O v . Then we shall say that v is a p-valuation [over K] [cf. Deﬁnition 2.2, (i)] if O K = O v ∩ K [which implies that p ∈ m v ]. Here, the phrase “over K” will be omitted in situations where the base ﬁeld K is ﬁxed throughout the discussion. We shall say that v is residue-transcendental def [cf. Deﬁnition 2.2, (i)] if it is a p-valuation whose residue ﬁeld k v = O v /m v is a transcendental extension of the residue ﬁeld of K. Let Z be a hyperbolic curve over K. Then we shall say that Z satisﬁes Σ-RNS [i.e., “Σ-resolution of nonsingularities” — cf. Deﬁnition 2.2, (vii); [Lpg1], Deﬁnition 2.1] if the following condition holds: Let v be a discrete residue-transcendental p-valuation on the func- tion ﬁeld K(Z) of Z. Then there exists a connected geometrically pro-Σ ﬁnite étale Galois covering Y → Z such that Y has stable reduction [over its base ﬁeld], and v coincides with the restriction [to K(Z)] of a discrete valuation on the function ﬁeld K(Y ) of Y that arises from an irreducible component of the special ﬁber of the stable model [cf. Deﬁnition 2.1, (vi)] of Y . If Z is an O K -scheme, then we shall write Z s for the special ﬁber of Z [i.e., the ﬁber of Z over the closed point of Spec O K ]. Let Z be an O K -scheme. Then [cf. Deﬁnition 2.1, (i), (ii), (iv)]: (i) We shall say that Z is a compactiﬁed model of Z over O K if Z is a proper, ﬂat, normal scheme over O K whose generic ﬁber is the [uniquely deter- mined, up to unique isomorphism] smooth compactiﬁcation of Z over K. (ii) Suppose that the cusps of Z are K-rational. Then we shall say that Z is a compactiﬁed stable model of Z over O K if Z is a compactiﬁed model of Z over O K such that the following conditions hold: • the geometric special ﬁber of Z is a semistable curve [i.e., a reduced, connected curve each of whose nonsmooth points is an ordinary dou- ble point]; • the images of the sections Spec O K → Z determined by the cusps of Z [which we shall refer to as cusps of Z] lie in the smooth locus of Z and do not intersect each other. • Z, together with the cusps of Z, determines a pointed stable curve. Then one veriﬁes immediately, by considering blow-ups of compactiﬁed models at speciﬁed closed points [cf. also Remark 2.2.2; Proposition 2.4, (iii), (iv)], that, if Σ is a set of cardinality ≥ 2 that contains p [cf. the situation discussed 3 in Theorem A below], then the condition of satisfying Σ-RNS discussed above is in fact equivalent to the following property, which clariﬁes the meaning of “resolution of nonsingularities”: Let Z be a compactiﬁed model of Z over O K ; z ∈ Z s a closed point. Then, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exist • a connected geometrically pro-Σ ﬁnite étale Galois covering Y → Z of hyperbolic curves over K, • a compactiﬁed stable model Y of Y over O K , • a morphism Y → Z of compactiﬁed models over O K that re- stricts to the ﬁnite étale Galois covering Y → Z, • an irreducible component D of Y s whose normalization is of genus ≥ 1, and whose image in Z s is z ∈ Z s . This alternative formulation of the condition of satisfying Σ-RNS is useful to keep in mind when considering the relationship between Theorem A below and the following result due to A. Tamagawa [cf. [Tama2]], which played an important role in motivating the following result due to E. Lepage [cf. [Lpg1]]: • Suppose that Σ = Primes, that the residue ﬁeld of K is algebraic over the ﬁnite ﬁeld of cardinality p, and that Z is the compactiﬁed stable model of Z over O K . Then there exist a connected ﬁnite étale Galois covering Y → Z and an irreducible component D of Y s as in the above alternative formulation [cf. [Tama2], Theorem 0.2, (v)]. • Suppose that Σ = Primes, that K is a p-adic local ﬁeld, and that Z is a hyperbolic Mumford curve over K. Then Z satisﬁes Σ-RNS [cf. [Lpg1], Theorem 2.7]. Our ﬁrst main result may be regarded as a generalization of these results [cf. Theorem 2.17]: Theorem A (Resolution of nonsingularities for arbitrary hyperbolic curves over p-adic local ﬁelds). Suppose that Σ ⊆ Primes is a subset of cardinality ≥ 2 that contains p, and that K is a p-adic local ﬁeld. Let X be a hyperbolic curve over K; L a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p that contains K as a topological subﬁeld. Then X L satisﬁes Σ-RNS if and only if the residue ﬁeld of L is algebraic over the ﬁnite ﬁeld of cardinality p. In the remainder of the present Introduction, we discuss various anabelian applications/consequences of Theorem A. First, by applying a certain sophisticated version of the argument applied in the proof of [Tsjm], Theorem 2.2, we obtain the following consequence of Theorem A concerning the determination of closed points on arbitrary hyperbolic curves via geometric tempered fundamental groups, which generalizes [Tsjm], Theorem 2.2 [cf. Corollary 2.5, (ii), which in fact applies to hyperbolic curves over more general p-adic ﬁelds; Remark 2.5.1]: 4 Theorem B (Determination of closed points on arbitrary p-adic hy- perbolic curves by geometric tempered fundamental groups). Suppose that Σ ⊆ Primes is a subset of cardinality ≥ 2 that contains p. Let X † , X ‡ † → X † , X ‡ → X ‡ for the univer- be hyperbolic curves over Q p . Write X , Π tp , respectively. Let sal geometrically pro-Σ coverings corresponding to Π tp X † X ‡ x † ∈ X † (C p ), x ‡ ∈ X ‡ (C p ). Write X x † † (respectively, X x ‡ ‡ ) for the hyperbolic curve X C † p \{x † } (respectively, X C ‡ p \{x ‡ }) over C p . Let ∼ σ : Π tp † → Π tp ‡ x x X † X ‡ be an isomorphism of topological groups that ﬁts into a commutative diagram ∼ Π tp † −−−−→ Π tp ‡ X † X ‡ σ ⏐ x ⏐ x ⏐ ⏐ ∼ Π tp −−−−→ Π tp , X † X ‡ σ where the vertical arrows are the natural surjections [determined up to com- position with an inner automorphism] induced by the natural open immersions X x † † → X C † p , X x ‡ ‡ → X C ‡ p of hyperbolic curves; the lower horizontal arrow σ is the isomorphism of topological groups [determined up to composition with an inner automorphism] induced by a(n) [uniquely determined] isomorphism ∼ σ X : X † → X ‡ of schemes over Q p . Then x ‡ = σ X (x † ). Next, we consider applications of Theorem A to Grothendieck Conjecture- type results in anabelian geometry. We begin by recalling the following question, which may may be considered as an absolute version of the Grothendieck Con- jecture for hyperbolic curves over p-adic local ﬁelds: Let X † , X ‡ be hyperbolic curves over p-adic local ﬁelds. Then is the natural map Isom(X † , X ‡ ) −→ OutIsom(Π X † , Π X ‡ ) bijective? This question may be regarded as one of the major open questions in anabelian geometry. In this context, we recall that, in the case of the relative version of the Grothendieck Conjecture for arbitrary hyperbolic curves, many satisfactory results have been obtained [cf. [PrfGC], Theorem A; [Tama1], Theorem 0.4; [LocAn], Theorem A]. In particular, the ﬁrst author of the present paper gave a complete aﬃrmative answer to the original question posed by A. Grothendieck [i.e., the original “Grothendieck Conjecture”] in quite substantial generality [cf. [LocAn], Theorem A; [AnabTop], Theorem 4.12]. On the other hand, in the case 5 of the absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local ﬁelds [i.e., the question of the above display], analogous results had not been obtained previously, due to the existence of outer isomor- phisms of the absolute Galois groups of p-adic local ﬁelds that do not arise from isomorphisms of ﬁelds [cf., e.g., [NSW], the Closing Remark preceding Theorem 12.2.7]. In this direction, in some sense the strongest known result, prior to the present paper, was the following result [cf. [AbsTopII], Corollary 2.9]: Suppose that Σ ⊆ Primes is a subset of cardinality ≥ 2 that contains p. Let X † , X ‡ be hyperbolic curves over p-adic local ﬁelds. Write (Σ) (Σ) (Σ) (Σ) OutIsom D (Π X † , Π X ‡ ) ⊆ OutIsom(Π X † , Π X ‡ ) for the subset determined by the isomorphisms that induce bijections between the respective sets of decomposition subgroups associated to the closed points of X † and X ‡ . Then the natural map Isom(X † , X ‡ ) −→ OutIsom D (Π X † , Π X ‡ ) (Σ) (Σ) is bijective. Moreover, in [AbsTopII], [AbsTopIII], the ﬁrst author developed a technique, called “Belyi cuspidalization”, that allows one to reconstruct the decomposition subgroups associated to the closed points of strictly Belyi-type hyperbolic curves and proved that an absolute version of the Grothendieck Conjecture holds for such curves [cf. [AbsTopIII], Theorem 1.9]. In the present paper, we apply Theorem A, together with some combinatorial anabelian geometry, to reconstruct the set of C p -valued points of a hyperbolic curve over Q p from its geometric tempered fundamental group [cf. Corollary 3.10, which in fact applies to hyperbolic curves over more general p-adic ﬁelds]: Theorem C (Reconstruction of C p -valued points via geometric tem- pered fundamental groups). Suppose that Σ ⊆ Primes is a subset of cardi- → X nality ≥ 2 that contains p. Let X be a hyperbolic curve over Q p . Write X tp may be for the universal pro-Σ covering corresponding to Π X [so Gal( X/X) tp identiﬁed with the pro-Σ completion of Π X ]. Then the set X(C p ) equipped — hence also, by passing to the set of with its natural action by Gal( X/X) p ) X(C p ) — may be reconstructed, Gal( X/X)-orbits, the quotient set X(C in a purely combinatorial/group-theoretic way and functorially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp X . Note that it follows immediately from Theorem C that, under the assump- tion that Σ ⊆ Primes is a subset of cardinality ≥ 2 that contains p, one may reconstruct the set of closed points, hence also the set of associated de- composition subgroups, of a hyperbolic curve over a p-adic local ﬁeld, in a purely combinatorial/group-theoretic way and functorially with respect to iso- morphisms of topological groups, from its geometrically pro-Σ étale fundamental group [cf. the proof of Theorem 3.11 for more details]. Here, it is also interesting to observe that 6 this reconstruction of the set of decomposition subgroups asso- ciated to closed points from the geometrically pro-Σ étale funda- mental group of a hyperbolic curve over a p-adic local ﬁeld may be thought of as a sort of a weak version of the Section Conjecture. In particular, by applying [AbsTopII], Corollary 2.9, together with some com- binatorial anabelian geometry, we obtain a complete aﬃrmative answer to the question considered above, i.e., an absolute version of the Grothendieck Con- jecture for arbitrary hyperbolic curves, as well as for the conﬁguration spaces associated to such hyperbolic curves, over p-adic local ﬁelds [cf. Theorems 3.12; 3.13]: Theorem D (Absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local ﬁelds). Suppose that Σ ⊆ Primes is a subset of cardinality ≥ 2 that contains p. Let X † , X ‡ be hyperbolic curves over p-adic local ﬁelds. Then the natural map Isom(X † , X ‡ ) −→ OutIsom(Π X † , Π X ‡ ) (Σ) (Σ) is bijective. Theorem E (Absolute version of the Grothendieck Conjecture for con- ﬁguration spaces associated to arbitrary hyperbolic curves over p-adic local ﬁelds). Let X † , X ‡ be hyperbolic curves over p-adic local ﬁelds; n † , n ‡ positive integers. Write X n † † (respectively, X n ‡ ‡ ) for the n † -th (respectively, n ‡ - th) conﬁguration space associated to X † (respectively, X ‡ ). Then the natural map Isom(X n † † , X n ‡ ‡ ) −→ OutIsom(Π X † , Π X ‡ ) n † n ‡ is bijective. In the context of the relationship between the theory developed in the present paper and the Section Conjecture, it is of interest to note the following conse- quences of the theory underlying Theorem C [cf. Proposition 2.4, (vii); Propo- sition 3.5, (iii); Proposition 3.9, (iv) — all of which in fact apply to hyperbolic curves over more general p-adic ﬁelds]: Theorem F (Consequences related to the Section Conjecture over p-adic ﬁelds). Suppose that Σ ⊆ Primes is a subset that contains p, and that the residue ﬁeld of K is an algebraic extension of the ﬁnite ﬁeld of cardinality p. Let l ∈ Σ \ {p}; H ⊆ G K a closed subgroup such that the intersection H ∩ I K of H with the inertia subgroup I K of G K admits a surjection to [the proﬁnite group] Z l . Write Ω for the p-adic completion of K; Ω H ⊆ Ω for the subﬁeld of Ω ﬁxed by H. Then the following properties hold: (i) Let X be a proper hyperbolic curve over K that is in fact deﬁned over a p-adic local ﬁeld contained [in a fashion compatible with the respective → X a universal geometrically pro-Σ covering, s X : topologies] in K, X 7 (Σ) def H → Π X = Gal( X/X) a section of the restriction to H of the natural (Σ) an for the topological pro-Berkovich space surjection Π X G K . Write X associated to [i.e., the inverse limit of the underlying topological spaces Then of the Berkovich spaces associated to the ﬁnite subcoverings of ] X. an there exists at most one point ∈ X that is ﬁxed by the restriction, via (Σ) s X , to H of the natural action of Π X on the topological pro-Berkovich an ; if, moreover, the restriction to H of the l-adic cyclotomic space X character of K has open image, then there exists a unique such point an . Finally, s X arises from an Ω H -rational point ∈ X(Ω H ) if and ∈ X (Σ) an . only if its image is a maximal stabilizer ⊆ Π X × G K H of a point ∈ X (ii) Suppose that the restriction to H of the l-adic cyclotomic character of K has open image. Let Y , Z be [not necessarily proper!] hyperbolic curves over K that are in fact deﬁned over a p-adic local ﬁeld contained [in a fashion compatible with the respective topologies] in K; Y → Y , Z → Z universal geometrically pro-Σ coverings of Y , Z, respectively; (Σ) def f : Y → Z a dominant morphism over K; s Y : H → Π Y = Gal( Y /Y ), (Σ) def = Gal( X/X) sections of the restrictions to H of the s Z : H → Π Z (Σ) (Σ) respective natural surjections Π Y G K , Π Z G K such that s Y is (Σ) mapped, up to Π Z -conjugation, by f to s Z via the map induced by f on geometrically pro-Σ fundamental groups. Then s Y arises from an Ω H - rational point ∈ Y (Ω H ) if and only if s Z arises from an Ω H -rational point ∈ Z(Ω H ). Here, we observe in passing that it follows immediately from Theorem F, (ii) [i.e., as stated above], together with [BSC], Theorem 5.33, that a similar result to the result stated in Theorem F, (ii), holds if the ﬁeld K is replaced by a number ﬁeld, but [since global issues over number ﬁelds lie beyond the scope of the present paper] we shall not discuss this in detail in the present paper. Next, we recall that one of the key ingredients in the theory of p-adic arith- metic cuspidalizations developed in [Tsjm], §2, is Lepage’s resolution of nonsin- gularities [i.e., [Lpg1], Theorem 2.7], which may be regarded as a special case of Theorem A. Our next result is obtained by applying this theory of p-adic arith- metic cuspidalizations [i.e., [Tsjm], §2], together with some elementary observa- tions concerning the lengths of nodes of stable models of hyperbolic curves [cf. Proposition 3.15], to prove that the various p-adic versions of the Grothendieck- Teichmüller group that appear in the literature [cf. [Tsjm], Remark 2.1.2] in fact coincide and are commensurably terminal in the Grothendieck-Teichmüller group [cf. Theorem 3.16; Corollary 3.17]: Theorem G (Equality and commensurable terminality of various p-adic versions of the Grothendieck-Teichmüller group). Suppose that Σ = def Primes. Write X = P 1 Q \ {0, 1, ∞}; p GT ⊆ Out(Π X ) 8 for the Grothendieck-Teichmüller group [cf. [CmbCsp], Remark 1.11.1]; GT M ⊆ GT (⊆ Out(Π X )) for the metrized Grothendieck-Teichmüller group [cf. [CbTpIII], Remark 3.19.2]; def = GT ∩ Out(Π tp GT tp p X ) ⊆ Out(Π X ) [cf. the subsection in Notations and Conventions entitled “Fundamental groups”; [Tsjm], Deﬁnition 2.1]. Then the natural inclusion GT M ⊆ GT tp p of subgroups of GT is an equality. In particular, it holds that GT M = GT p = GT G = GT tp p [cf. [Tsjm], Remark 2.1.2]. Moreover, GT M = GT p = GT G = GT tp p is com- M mensurably terminal in GT, i.e., the commensurator C GT (GT ) of GT M in GT is equal to GT M . Note that the commensurable terminality in the ﬁnal portion of Theorem G may be regarded as an aﬃrmative answer to the question posed in the discussion immediately preceding Theorem E in [CbTpIII], Introduction. Finally, we conclude with an interesting complement to the theory of p-adic arithmetic cuspidalizations by applying Theorem C, together with the theory of metric-admissibility developed in [CbTpIII], §3, to construct certain p-adic arithmetic cuspidalizations of the geometric tempered fundamental group of a hyperbolic curve over Q p equipped with certain relatively mild auxiliary data [cf. Deﬁnition 3.19; Theorem 3.20; Remarks 3.20.1, 3.20.2]. The contents of the present paper may be summarized as follows: In §1, we discuss in detail certain local computations — motivated by [Lpg1], Proposition 2.4, but formulated entirely in the language of schemes and for- mal schemes, i.e., without resorting to the use of notions from the theory of Berkovich spaces — concerning iterates of the p-th power morphism of the mul- tiplicative group scheme G m over the ring of integers of a mixed characteristic discrete valuation ﬁeld of residue characteristic p. By restricting such mor- phisms to suitable formal neighborhoods, we conclude that smooth curves of genus ≥ 1 appear in the special ﬁbers of suitable models of the domain curves of such morphisms, i.e., as certain Artin-Schreier coverings of curves of genus 0 [cf. Proposition 1.6; Remark 1.6.2]. In §2, we begin by discussing various generalities concerning models of a hyperbolic curve over a mixed characteristic complete discrete valuation ﬁeld [cf. Deﬁnition 2.1; Proposition 2.3]. In particular, we discuss the deﬁnition of the notion of Σ-RNS [cf. Deﬁnition 2.2, (vii)], together with closely related basic properties of this notion [cf. Propositions 2.4; Corollary 2.5]. Another important 9 notion in this context is the purely combinatorial notion of VE-chains associated to a hyperbolic curve over a mixed characteristic complete discrete valuation ﬁeld [cf. Deﬁnition 2.2, (iii)]. This notion is closely related to the topological Berkovich space associated to the hyperbolic curve [cf. Proposition 2.3, (vii), (viii)]. Finally, we apply certain constructions involving p-divisible groups to extend the Artin-Schreier coverings constructed locally in §1 to coverings of an arbitrary hyperbolic curve over a p-adic local ﬁeld [cf. Propositions 2.12, 2.13; Theorem 2.16]. This leads naturally to a proof of Theorem A [cf. Theorem 2.17], hence also, by combining Theorem A with the theory of VE-chains developed in the earlier portion of §2, together with some combinatorial anabelian geometry, of Theorem B. In §3, we begin by recalling the well-known classiﬁcation of the points of the topological Berkovich space associated to a proper hyperbolic curve over a mixed characteristic complete discrete valuation ﬁeld via the notion of type i points, where i ∈ {1, 2, 3, 4}. Next, we introduce a certain combinatorial clas- siﬁcation of the VE-chains considered in §2 and observe that this classiﬁcation of VE-chains leads naturally to a purely combinatorial characterization of the well-known classiﬁcation via type i points mentioned above. We then apply the theory of §2 to give a group-theoretic characterization, motivated by [but by no means identical to] the characterization of [Lpg2], §4, of the type i points in terms of the geometric Σ-tempered fundamental group of the hyperbolic curve. The theory surrounding this group-theoretic characterization leads naturally to proofs of Theorems C and F. Moreover, by combining this group-theoretic characterization with [AbsTopII], Corollary 2.9; [HMM], Theorem A, we obtain proofs of Theorems D and E [cf. Theorems 3.12; 3.13]. We then switch gears to discuss metric-admissibility for p-adic hyperbolic curves. This discussion of metric-admissibility yields, in particular, a proof of Theorem G and leads naturally to the discussion of p-adic arithmetic cuspidalizations associated to geometric tempered fundamental groups equipped with certain relatively mild auxiliary data [cf. Deﬁnition 3.19; Theorem 3.20; Remarks 3.20.1, 3.20.2] men- tioned above. Acknowledgements This research was supported by the Research Institute for Mathematical Sci- ences, an International Joint Usage/Research Center located in Kyoto Univer- sity, as well as the International Center for Research in Next Generation Geom- etry. 10 Notations and Conventions Numbers: The notation Primes will be used to denote the set of prime numbers. The notation N will be used to denote the set of nonnegative integers. The notation Q ≥0 will be used to denote the additive monoid of nonnegative rational numbers. The notation Z will be used to denote the additive group or ring of integers. The notation Q will be used to denote the ﬁeld of rational numbers. The notation R will be used to denote the ﬁeld of real numbers. For each x ∈ R, the notation x will be used to denote the greatest integer ≤ x. If p is a prime number, then the notation Q p will be used to denote the ﬁeld of p-adic numbers; the notation Z p will be used to denote the additive group or ring of p-adic integers; the notation Q p will be used to denote an algebraic closure of Q p ; the notation C p will be used to denote the p-adic completion of Q p . We shall refer to a ﬁnite extension ﬁeld of Q p as a p-adic local ﬁeld. Monoids: Let M be a commutative monoid. In this subsection, we regard the set of positive integers N ≥1 as a directed set via its multiplicative structure, i.e., def i ≤ j ⇔ i | j. For i ∈ N ≥1 , write M i = M . For i, j ∈ N ≥1 such that i ≤ j, write φ i,j : M i → M j for the homomorphism of monoids obtained by multiplication by ji . Then we shall refer to the inductive limit of the inductive system {M i , φ i,j } on the directed set N ≥1 as the perfection of M . Rings and ﬁelds: Let R be a ring. Then we shall write R × for the multiplicative group of units of the ring. Let F be a perfect ﬁeld, p a prime number. Then the notation F will be def used to denote an algebraic closure of F ; G F = Gal(F /F ). Suppose that F is a valuation ﬁeld, i.e., a ﬁeld equipped with a valuation map [cf., e.g., the axioms of [EP], Eq. (2.1.2)]. Then we shall write O F for the ring of integers of F ; m F for the maximal ideal of O F . Thus, the valuation map on F induces an isomorphism of ordered abelian groups between F × /O F × and the value group of the valuation ﬁeld F . In particular, any valuation map on the ﬁeld F is determined, up to unique isomorphism [in the evident sense], by the ring of integers ⊆ F associated to the valuation map. In the present paper, we shall use the term valuation to refer to an isomorphism class of valuation maps on a ﬁeld F , i.e., the collection of valuation maps that give rise to the same ring of integers ⊆ F . We shall refer to a speciﬁc valuation map within a given isomorphism class of valuation maps on a ﬁeld as a normalized valuation, i.e., an isomorphism class of valuation maps on the ﬁeld, together with a speciﬁc valuation map [belonging to this class], which we shall refer to as a normalization. If the value group of the 11 valuation ﬁeld F is isomorphic, as an ordered abelian group, to a subgroup of the underlying additive group of R, then we shall say that F is a real valuation ﬁeld. If F is a mixed characteristic real valuation ﬁeld of residue characteristic p, then the notation v p will be used to denote the normalized valuation on F whose normalization is determined by the condition that v p (p) = 1 ∈ R. If F is a henselian valuation ﬁeld of characteristic 0, then we shall write I F ⊆ G F for the inertia subgroup; F ⊆ F ur (⊆ F ) for the maximal unramiﬁed extension. If F is a real henselian valuation ﬁeld of characteristic 0, then we shall write F ur for the completion of F ur [cf. Remark 2.2.4 for more details]. Topological groups: Let G be a topological group; H ⊆ G a closed subgroup of G. Then we shall write G ab for the abelianization of G; C G (H) for the commensurator of H ⊆ G, i.e., def C G (H) = {g ∈ G | H ∩ g · H · g −1 is of ﬁnite index in H and g · H · g −1 }. We shall say that the closed subgroup H is commensurably terminal in G if H = C G (H). Let Σ ⊆ Primes be a nonempty subset. Then we shall write G Σ for the pro-Σ completion of G. We shall write Aut(G) for the group of continuous automorphisms of the topological group G, Inn(G) ⊆ Aut(G) for the subgroup of inner automorphisms def of G, and Out(G) = Aut(G)/Inn(G). Suppose that G is center-free. Then we have a natural exact sequence of groups ∼ 1 −→ G ( → Inn(G)) −→ Aut(G) −→ Out(G) −→ 1. If J is a group, and ρ : J → Out(G) is a homomorphism, then we shall denote by out G J the group obtained by pulling back the above exact sequence of groups via ρ. Thus, we have a natural exact sequence of groups out 1 −→ G −→ G J −→ J −→ 1. Suppose further that the topology of G admits a countable basis consisting of characteristic open subgroups of G. Then one veriﬁes immediately that the topology of G induces a natural topology on the group Aut(G), hence on the group Out(G). In particular, one veriﬁes easily that if J is a topological group, out and ρ : J → Out(G) is continuous, then G J admits a natural topological group structure. Let G 1 , G 2 be proﬁnite groups. Then we shall write OutIsom(G 1 , G 2 ) 12 for the set of isomorphisms of proﬁnite groups, considered up to composition with an inner automorphism arising from an element ∈ G 2 . Semi-graphs: Let Γ be a connected semi-graph [cf. [SemiAn], §1]. Then we shall refer to the dimension over Q of the ﬁrst homology module of Γ [with coeﬃcients in Q] H 1 (Γ, Q) as the loop-rank of Γ. We shall say that Γ is untangled if every closed edge of Γ abuts to two distinct vertices. Schemes: Let K be a ﬁeld; K ⊆ L a ﬁeld extension; X an algebraic variety [i.e., a separated, geometrically integral scheme of ﬁnite type] over K. Then we shall def write X L = X × K L. Suppose that X is a smooth proper curve [i.e., a smooth, proper algebraic variety of dimension 1] over K. Then we shall write J(X) for the Jacobian of X. Let p be a prime number; A a semi-abelian variety or a p-divisible group over K. Then we shall write T p A for the p-adic Tate module associated to [the p-power torsion points valued in some ﬁxed algebraic closure of K of] A. Suppose that K is a valuation ﬁeld. Let X be an O K -scheme. Then we shall write X s for the special ﬁber of X [i.e., the ﬁber of X over the closed point of Spec O K ]. Let S 1 , S 2 be schemes. Then we shall write Isom(S 1 , S 2 ) for the set of isomorphisms of schemes between S 1 and S 2 . Curves: We shall use the term “hyperbolic curve” [i.e., a family of hyperbolic curves over the spectrum of a ﬁeld] as deﬁned in [MT], §0. We shall use the term “n-th conﬁguration space” as deﬁned in [MT], Deﬁnition 2.1, (i). Log schemes: If X log is a ﬁne log scheme, then we shall write • X for the underlying scheme of X log ; • M X for the étale sheaf of monoids on X that deﬁnes the log structure of X log ; × , which we shall refer to as the characteristic of X log [cf. • M X = M X /O X [MT], Deﬁnition 5.1, (i)]. def Let K be a complete discrete valuation ﬁeld; Y a hyperbolic curve over K; Y a def compactiﬁed semistable model of Y over O K [cf. Deﬁnition 2.1, (ii)]. Write S = Spec O K ; S log for the log scheme determined by the log structure associated to the closed point of S. Then it follows immediately from [Hur], §3.7, §3.8, 13 that the multiplicative monoid of sections of O Y that are invertible on [the open subscheme of Y determined by] Y determines a natural log structure on Y. Denote the resulting log scheme by Y log . Then one veriﬁes immediately that the natural morphism of schemes Y → S extends to a a proper, log smooth morphism Y log → S log of ﬁne log schemes. Let y be a geometric point of Y s . Write M y pf for the perfection of the stalk of the characteristic M Y at y. Then one veriﬁes immediately that, if the image of y in Y s is a smooth point that is not a cusp (respectively, a cusp; a node), then ∼ ∼ ∼ M y pf → Q ≥0 (respectively, M y pf → Q ≥0 × Q ≥0 ; M y pf → Q ≥0 × Q ≥0 ). Fundamental groups: For a connected noetherian scheme S, we shall write Π S for the étale funda- mental group of S, relative to a suitable choice of basepoint. Let Σ ⊆ Primes be a nonempty subset; K a perfect ﬁeld; X an algebraic variety over K. Then we shall write def Δ X = Π X K ; (Σ) def Π X = Π X /Ker(Δ X Δ Σ X ), Σ where Δ X Δ Σ X denotes the natural surjection. We shall refer to Δ X (respec- (Σ) tively, Π X ) as the geometric pro-Σ fundamental group (respectively, geometri- cally pro-Σ fundamental group) of X. Let p be a prime number, Σ ⊆ Primes a nonempty subset. Suppose that K is a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p. Write Ω for the p-adic completion of K. Let F be one of the [topological] ﬁelds K, K, and Ω; X a hyperbolic curve over F . If F = Ω, then we shall write Π tp X for the Σ-tempered fundamental group of X, i.e., in the terminology of [CbTpIII], Deﬁnition 3.1, (ii), the “Σ-tempered fundamental group” of the smooth log curve over F determined by X [where we note that one veriﬁes immediately that the ∧ ﬁeld “ K ” of loc. cit. may be taken to to be the ﬁeld F = Ω of the present discussion, and that the discussion of loc. cit. may be applied to the situation of the present discussion even if the X of the present discussion does not descend to the ﬁeld “ K” of loc. cit.]. Here, we recall that, when F = Ω, it follows immediately from [André], Proposition 4.3.1 [and the surrounding discussion], that the universal topological coverings of ﬁnite coverings corresponding to char- acteristic open subgroups of Π tp X of ﬁnite index determine a countable collection tp of characteristic open subgroups of Π tp X that form a basis of the topology of Π X , tp hence determine a natural topology on Out(Π X ). Thus, if F = K, then we obtain a natural continuous outer action G K Out(Π tp X Ω ) −→ 14 and hence, since Π tp X Ω is center-free [cf., e.g., [CbTpIII], Proposition 3.3, (i), (ii); [MT], Proposition 1.4], a topological group Π tp X def = out Π tp X Ω G K , which we refer to as the geometrically Σ-tempered fundamental group of X. Moreover, if F = K, then Π tp X is equipped with a natural continuous surjection tp Π X G K , whose kernel [which in fact may be naturally identiﬁed with Π tp X Ω ] we def = Π tp denote by Δ Σ-tp X X Ω and refer to as the geometric Σ-tempered fundamental group of X. If F = K, then, after possibly replacing K by a ﬁnite extension of K, we may assume that X descends to a hyperbolic curve X K over K [so that X may be naturally identiﬁed with “(X K ) K ”], and we shall refer to Π tp X def = Π tp X Ω as the Σ-tempered fundamental group of X. Finally, we note that, in the case where F = K, it follows immediately from [CanLift], Proposition 2.3, (ii) [cf. also [CbTpIII], Proposition 3.3, (i)], together with the deﬁnition of the Σ- tp tempered fundamental group, that the pro-Σ completion of Π tp X = Π X Ω may be naturally identiﬁed, for suitable choices of basepoints, with the geometric pro-Σ fundamental group Δ Σ X of X. 1 Local construction of Artin-Schreier extensions in the special ﬁber Let p be a prime number. In the present section, we perform various local computations concerning iterates of the p-th power morphism of the multiplica- tive group scheme G m over the ring of integers of a mixed characteristic discrete valuation ﬁeld of residue characteristic p. As a consequence, by restricting such morphisms to suitable formal neighborhoods, we conclude that smooth curves of genus ≥ 1 appear in the special ﬁbers of suitable models of the domain curves of such morphisms [cf. Proposition 1.6; Remark 1.6.2]. This observation will be applied in §2 to prove that arbitrary hyperbolic curves over p-adic local ﬁelds ad- mit resolution of nonsingularities. The contents of this section may be regarded as an alternative and somewhat more detailed discussion of [Lpg1], Proposition 2.4, that is phrased entirely in the language of schemes and formal schemes and does not resort to the use of Berkovich spaces. First, we begin with several elementary lemmas concerning the p-adic valu- ations of the coeﬃcients of certain polynomials and power series [cf. Lemmas 1.1, 1.2, 1.3]. 15 Lemma 1.1. Let K be a mixed characteristic discrete valuation ﬁeld of residue characteristic p. Suppose that K contains a primitive p-th root of unity ζ p ∈ K. def Write π = 1 − ζ p ∈ K; def f (x) = π −p (1 + πx) p − 1 ∈ K[x], where x denotes an indeterminate. Then it holds that f (x) − (x p − x) ∈ m K [x] ⊆ O K [x]. def Proof. First, write c = p + (−π) p−1 . Then since π = 1 − ζ p ∈ K, it holds that c = p + (−π) p−1 = p + (−π) p−1 + π −1 ((1 − π) p − 1). Observe that these equalities imply that v p (c) > 1. In particular, it holds that v p (p + (−π) p−1 ) > 1, hence that v p (p + π p−1 ) > 1. Next, observe that, if we write f (x) = a i x i ∈ K[x], 1≤i≤p then since v p (p) = 1 = v p (π p−1 ), and v p (p + π p−1 ) > 1, it holds that a 1 = p , π p−1 a p = 1, v p (a 1 + 1) > 0, v p (a i ) > 0 (∀i = 1, p). Thus, we conclude that f (x) − (x p − x) ∈ m K [x]. This completes the proof of Lemma 1.1. Lemma 1.2. Let K be a mixed characteristic discrete valuation ﬁeld of residue characteristic p; n a positive integer. Write q i x i ∈ K[[x]] f (x) = 1 + i≥1 — where x denotes an indeterminate — for the n-th root of 1+x ∈ K[[x]] whose constant term is equal to 1. Then it holds that v p (q 1 ) = −v p (n), v p (q i ) ≥ −iv p (n) − v p (i!) ≥ −i v p (n) + 1 p − 1 (∀i ≥ 2) i [cf. the well-known elementary fact that v p (i!) ≤ p−1 ]. If, moreover, n is prime to p, then v p (q i ) ≥ 0 for each positive integer i. 1 Proof. First, we observe, by considering the Taylor expansion of (1 + x) n at 0, that 1 1 − k q i = i! n 0≤k≤i−1 16 for each positive integer i. The equality and inequalities of the second display of the statement of Lemma 1.2 follow immediately. Next, suppose that n is prime to p. For each positive integer m, write def Q m ⊆ K for the O K -subalgebra generated by {q j } 1≤j≤m−1 . Write Q 0 = O K . Then, by comparing the coeﬃcients of x m in the left- and right-hand sides of the equality n 1+ q i x i = 1 + x, i≥1 we conclude that nq m ∈ Q m−1 . In particular, since n is prime to p, it holds that q m ∈ Q m−1 . Thus, by induction, we conclude that q i ∈ O K for each positive integer i. This completes the proof of Lemma 1.2. Lemma 1.3. Let K be a mixed characteristic discrete valuation ﬁeld of residue characteristic p; n a positive integer; a i s i ∈ O K [[s]] \ {1}, g(s) = 1 + i≥1 where s denotes an indeterminate. Write i 0 for the smallest positive integer i such that a i = 0; 0 s for the O K -valued point of Spec O K [[s]] obtained by mapping s → 0. Then the following hold: (i) Suppose that v p (a i ) ≥ 2v p (n) for each i ≥ 1. Then g(s) admits an n-th root 1+ b i s i ∈ O K [[s]]. i≥1 1 (ii) Suppose that there exists an element x ∈ O K such that v p (x) = 3i 0 (p−1) . Then there exist • a positive integer j, 2 ≤ • an element b ∈ O K satisfying the inequalities v p (b) ≥ 1 and 3(p−1) 1 v p (b) − v p (b) < p−1 , and ∼ • an isomorphism h : O K [[s]] → O K [[s]] of topological O K -algebras such that h maps 0 s → 0 s and h(g(x j s)) = 1 + bs i 0 . (iii) Suppose that g(s) = 1+a i 0 s i 0 , where a i 0 satisﬁes the inequalities v p (a i 0 ) ≥ def 1 2 1 and 3(p−1) ≤ v p (a i 0 ) − v p (a i 0 ) < p−1 . Write μ = v p (a i 0 ) − 1 ≥ 0. μ Then g(s) admits a p -th root 1 c i 0 i s i 0 i ∈ O K [[s]], g(s) pμ = 1 + i≥1 p 2 2 ≤ v p (c i 0 ) < p−1 , v p (c 2i 0 ) ≥ 2 1 + 3(p−1) − 1, and where 1 + 3(p−1) 1 v p (c i 0 i ) ≥ i 1 − 3(p−1) for each positive integer i > 2. 17 (iv) In the notation of (iii), suppose that K contains a primitive p-th root of unity ζ p ∈ K and an element c ∈ K such that c i 0 = π p , c i 0 def p where we write π = 1 − ζ p . [Note that since v p (c i 0 ) < p−1 , it holds that v p (c) > 0.] Write 1 d i 0 i s i 0 i = π −p (g(cs) pm − 1) ∈ K[[s]]. def i≥1 p , v p (d 2i 0 ) > Then it holds that d i 0 = 1, v p (d i 0 ) = 0 > v p (c i 0 ) − p−1 p 2i 0 sup{v p (c ) − p−1 , 0}, and v p (d i 0 i ) > i · sup 1 − 1 p , v p (c i 0 ) − ≥ 0 3(p − 1) p − 1 for each positive integer i > 2. Moreover, v p (d i 0 i ) > v p (c i 0 i ) for each suﬃciently large positive integer i. Proof. First, we verify assertion (i). Note that g(s) admits an n-th root 1+ b i s i ∈ K[[s]]. i≥1 Thus, it suﬃces to verify that v p (b i ) ≥ 0 for each positive integer i. Note that 1 if v p (n) ≥ 1 (≥ p−1 ), then 2iv p (n) − i v p (n) + 1 p − 1 ≥ 0. Thus, by applying Lemma 1.2 [where we take “x” to be the element i≥1 a i s i ], together with our assumption that v p (a i ) ≥ 2v p (n) for each positive integer i, we conclude that v p (b i ) ≥ 0 for each positive integer i. This completes the proof of assertion (i). Next, we verify assertion (ii). Fix an element x ∈ O K such that v p (x) = def ij 1 3i 0 (p−1) . For each pair of positive integers (i, j), write b i,j = x a i . Note that it follows immediately from our assumption on v p (x) [i.e., by thinking of the real 1 line R modulo integral multiples of 3(p−1) = i 0 v p (x)] that there exists a positive integer j such that • v p (b i 0 ,j ) ≥ 1, 2 1 • 3(p−1) ≤ v p (b i 0 ,j ) − v p (b i 0 ,j ) < p−1 , and 18 • v p (b i,j ) ≥ v p (b i 0 ,j ) + 2v p (i 0 ) for each positive integer i > i 0 . def Fix such a positive integer j and write b = b i 0 ,j . Then the existence of an ∼ isomorphism h : O K [[s]] → O K [[s]] of topological O K -algebras such that h maps 0 s → 0 s and h(g(x j s)) = 1 + bs i 0 follows immediately from Lemma 1.3, (i), where we take “n” to be i 0 and “g(s)” to be g † (s) = b −1 s −i 0 (g(x j s) − 1) def 1 def [so g(x j s) = 1+bs i 0 g † (s)]. [That is to say, h is deﬁned by taking h(s(g † (s)) n ) = 1 s, where “(g † (s)) n ” denotes the n-th root of Lemma 1.3, (i).] This completes the proof of assertion (ii). Assertion (iii) follows immediately from Lemma 1.2 [applied to x = a i 0 s i 0 ], together with the elementary fact that v p (i!) ≤ 1 for i = 2. Finally, we verify assertion (iv). It follows immediately from the various deﬁnitions involved that p(i−1) d i 0 i = c i 0 i c −i . i 0 π 1 In particular, it holds that d i 0 = 1. Moreover, since v p (π) = p−1 , and v p (c i 0 ) < p [cf. Lemma 1.3, (iii)], it holds that p−1 v p (d i 0 i ) = v p (c i 0 i ) − iv p (c i 0 ) + p(i − 1) p > v p (c i 0 i ) − . p − 1 p − 1 Suppose that i = 2 (respectively, i ≥ 3). Then it follows immediately from Lemma 1.3, (iii), that v p (c i 0 i ) − 2 p ≥ 2 1+ p − 1 3(p − 1) − 1 − p 1 = > 0 p − 1 3(p − 1) (respectively, v p (c i 0 i )− 1 1 2p − 4 p p p ≥ i 1− ≥ 3 1− = ≥ 0). − − p − 1 3(p − 1) p − 1 3(p − 1) p − 1 p − 1 Thus, we conclude that v p (d 2i 0 ) > 0, and v p (d i 0 i ) > i 1 − 1 3(p − 1) − p ≥ 0 p − 1 for each positive integer i > 2. The remainder of assertion (iv) follows immedi- ately from the inequalities already obtained, together with the inequalities p 1 1 p > inf 1 − , + p − 1 3(p − 1) 2 · 3(p − 1) 2(p − 1) p 2 1 = − 1+ > 3(p − 1) p − 1 3(p − 1) ≥ v p (c i 0 ) [cf. Lemma 1.3, (iii)]. This completes the proof of assertion (iv), hence of Lemma 1.3. 19 Deﬁnition 1.4. Let K be a mixed characteristic discrete valuation ﬁeld of residue characteristic p; c ∈ m K \ {0}. Then we shall write φ c : B c −→ Spec O K [[t]] — where t denotes an indeterminate — for the blow-up of Spec O K [[t]] with center given by the closed subscheme deﬁned by the ideal (c, t); b c ∈ B c for the generic point of the exceptional irreducible component of the special ﬁber of B c [i.e., the ﬁber of B c over the closed point of Spec O K ]; U c = Spec O K [[t]][s c ]/(cs c − t) ⊆ B c — where s c denotes an indeterminate, but, by a slight abuse of notation, we shall also use the notation “s c ” to denote the element of Γ(U c , O U c ) determined by the indeterminate “s c ”— for the open subscheme obtained by removing the strict transform of the special ﬁber of Spec O K [[t]]. Note that it follows immediately from the various deﬁnitions involved that φ c induces a morphism c def = Spec O K [[s c ]] −→ Spec O K [[t]] φ c : U over O K that maps t to cs c . Proposition 1.5 (Local construction of Artin-Schreier extensions in the special ﬁber I). We maintain the notation of Deﬁnition 1.4. Suppose that K contains a primitive p-th root of unity ζ p ∈ K. Write k for the residue def ﬁeld of K; π = 1 − ζ p ∈ K; f : Spec O K [[t]] −→ Spec O K [[t]] for the [manifestly] ﬁnite ﬂat morphism over O K corresponding to the homo- morphism of topological O K -algebras that maps t → (1+t) p −1. Then f induces a ﬁnite morphism f ˜ : B π −→ B π p over O K that maps b π → b π p and induces a ﬁnite ﬂat morphism U π → U π p , whose induced morphism on special ﬁbers is the morphism induced on spectra by the injective homomorphism k[s π p ] −→ k[s π ] over k that maps s π p → s pπ − s π . Proof. First, observe that we have inclusions of ideals in O K [[t]] 2 (π, t) p +p ⊆ (π p , (1 + t) p − 1) ⊆ (π, t). Indeed, the inclusion (π p , (1 + t) p − 1) ⊆ (π, t) is immediate. Next, we verify the 2 def inclusion (π, t) p +p ⊆ (π p , (1 + t) p − 1). Write t̃ = (1 + t) p − 1. Then it holds 20 2 2 2 that t̃ p ∈ t p + (p 2 ) ⊆ t p + (π p ). In particular, it holds that t p ∈ ( t̃ p , π p ) ⊆ 2 (π p , t̃). Thus, we conclude that π i t p +p−i ∈ (π p , t̃) for each integer i such that 2 0 ≤ i ≤ p 2 + p, hence that (π, t) p +p ⊆ (π p , (1 + t) p − 1). This completes the proof of the observation. The observation of the preceding paragraph, together with Lemma 1.1, im- plies, by the deﬁnition of the blow-up, that the morphism f functorially induces a [proper and quasi-ﬁnite, hence] ﬁnite morphism f ˜ : B π → B π p over O K , which, in turn, induces a [manifestly] ﬁnite ﬂat morphism U π −→ U π p over O K such that t → (1 + t) p − 1, s π p → π −p (1 + πs π ) p − 1 [cf. Lemma 1.1]. Indeed, it follows from Lemma 1.1 that this morphism U π → U π p determines a dominant morphism Spec k[s π ] −→ Spec k[s π p ] [between open subschemes of the respective exceptional irreducible components of the special ﬁbers of B π , B π p ] that corresponds to the injective homomorphism k[s π p ] −→ k[s π ] over k that maps s π p → s pπ − s π . This completes the proof of Proposition 1.5. Remark 1.5.1. In the notation of Proposition 1.5, write G m = Spec O K [u, u 1 ] for the multiplicative group scheme over O K , where u denotes an indeterminate; ι : Spec O K [[t]] −→ G m for the morphism that corresponds to the homomorphism over O K that maps u → 1 + t. Then we have a commutative diagram U π ⏐ ⏐ f ˜ | Uπ −−−−→ Spec O K [[t]] −−−−→ G m ι φ π | U π ⏐ ⏐ ⏐ ⏐ p f U π p −−−−−→ Spec O K [[t]] −−−−→ G m , φ π p | U π p ι where the right-hand vertical arrow denotes the p-th power morphism; the verti- cal arrows are ﬁnite ﬂat morphisms of degree p [cf. Proposition 1.5]; the second square is cartesian; the ﬁrst square is cartesian, up to taking the normalization of the ﬁber product that would make the ﬁrst square “truly cartesian”. 21 Proposition 1.6 (Local construction of Artin-Schreier extensions in the special ﬁber II). We maintain the notation of Remark 1.5.1. Let a i t i ∈ O K [[t]] \ {1}. g(t) = 1 + i≥1 Write i 0 for the smallest positive integer i such that a i = 0; λ g : Spec O K [[t]] −→ Spec O K [[t]] for the morphism over O K corresponding to the homomorphism of topological O K -algebras that maps t → g(t) − 1. Then the following hold: (i) After possibly replacing K by a suitable ﬁnite ﬁeld extension of K, there exist a positive integer μ, elements c 1 , c 2 ∈ m K \ {0}, an isomorphism ∼ λ h : Spec O K [[s c 1 ]] → Spec O K [[s c 1 ]] over O K , and a morphism ξ g : Spec O K [[s c 1 ]] −→ G m over O K satisfying the following conditions: • Write 0 c 1 for the O K -valued point of Spec O K [[s c 1 ]] obtained by map- ping s c 1 → 0. Then λ h maps 0 c 1 → 0 c 1 , and ξ g maps 0 c 1 to the identity element of G m (O K ). • There exists a commutative diagram Spec O K [[s c 1 ]] −−−−→ Spec O K [[t]] −−−−→ Spec O K [[t]] c φ 1 ⏐ λ h ⏐ λ g ⏐ ⏐ ι Spec O K [[s c 1 ]] −−−−→ ξ g G m −−− μ −→ p G m , where the right-hand lower horizontal arrow denotes the p μ -th power morphism. • Write ∼ τ : Spec O K [[t]] → Spec O K [[s c 1 ]] for the isomorphism over O K corresponding to the isomorphism of topological O K -algebras that maps s c 1 → t; η(s c 2 ) ∈ O K [[s c 2 ]] for the image of u via the homomorphism O K [u, u 1 ] → O K [[s c 2 ]] in- duced by the composite ∼ Spec O K [[s c 2 ]] −→ Spec O K [[t]] −→ Spec O K [[s c 1 ]] −→ G m . c φ 2 τ 22 ξ g Then it holds that η(s c 2 ) − 1 ∈ m K [[c 2 s c 2 ]], and, moreover, π −p (η(s c 2 ) − 1) − s i c 0 2 ∈ m K [s c 2 ] + m K [[c 2 s c 2 ]] = m K [s c 2 ] + m K [[t]]. In particular, there exists a morphism θ g : U c 2 −→ U π p over O K that ﬁts into the following commutative diagram ∼ U c 2 −−−−−→ Spec O K [[t]] −−−−→ Spec O K [[s c 1 ]] τ φ c 2 | U c 2 ⏐ ⏐ ⏐ ⏐ θ g ξ g U π p −−−−−→ Spec O K [[t]] −−−−→ φ π p | U π p ι G m . (ii) Fix a collection of data (μ, c 1 , c 2 , λ h , ξ g ) as in (i). Write def Y = U c 2 × U π p U π for the ﬁber product determined by the morphism θ g : U c 2 → U π p and the morphism U π → U π p induced by f ˜ [cf. Proposition 1.5]. Then the natural morphism Y s −→ (U c 2 ) s = Spec k[s c 2 ] induced by the ﬁrst projection morphism Y → U c 2 corresponds to the nat- ural injective homomorphism k[s c 2 ] → k[s c 2 , y]/(y p − y − s i c 0 2 ) over k, where y denotes an indeterminate. Proof. First, we consider assertion (i). We begin by applying Lemma 1.3, (ii), where we take “g(s)” to be g(t) [i.e., so the indeterminate “s” corresponds to t], and we observe that, by replacing K by a suitable ﬁnite extension of K, we may assume without loss of generality that there exists an “x” as in Lemma 1.3, ∼ (ii). This yields an isomorphism “h : O K [[s]] → O K [[s]]” as in Lemma 1.3, (ii), whose induced morphism on spectra — where we interpret the indeterminate “s” to be s c 1 , and we take “x j ” to be c 1 [so “x j s” corresponds to c 1 s c 1 = t] — we take to be λ h . Here, we recall that this isomorphism “h” of Lemma 1.3, (ii), satisﬁes a condition “h(g(x j s)) = 1 + bs i 0 ”. Next, we would like to apply Lemma 1.3, (iii), where we take “1 + a i 0 s i 0 ” to be the “1 + bs i 0 ” of Lemma 1.3, (ii) [i.e., so the indeterminate “s” still corresponds to s c 1 ]. This yields a 1 power series “g(s) pμ ” as in Lemma 1.3, (iii). We then take the “μ” of Lemma 1.3, (iii), to be μ and deﬁne ξ g to be the morphism over O K corresponding 1 to the homomorphism that maps u to this power series “g(s) pμ ” [i.e., where the indeterminate “s” still corresponds to s c 1 ]. This yields a collection of data 23 (μ, c 1 , λ h , ξ g ) that satisﬁes the ﬁrst two itemized conditions of Proposition 1.6, (i). The third [and ﬁnal] itemized condition of Proposition 1.6, (i), now follows by translating the various estimates of Lemma 1.3, (iv), into the notation of the present situation, where we take the “c” of Lemma 1.3, (iv), to be c 2 , and we observe that, again by replacing K by a suitable ﬁnite extension of K, we may assume without loss of generality that there exist “ζ p ” and “c” as in Lemma 1 1.3, (iv). Also, we observe that the power series “g(cs) pμ ” of Lemma 1.3, (iv), corresponds to η(s c 2 ) [i.e., where the indeterminate “s” corresponds to s c 2 ]. This completes the proof of assertion (i). Next, we verify assertion (ii). Recall that • the morphism θ g : U c 2 → U π p corresponds to the homomorphism of topo- logical O K -algebras O K [[t]][s π p ]/(π p s π p − t) −→ O K [[t]][s c 2 ]/(c 2 s c 2 − t) that maps s π p → π −p (η(s c 2 ) − 1), t → η(s c 2 ) − 1, while • the morphism U π → U π p corresponds to the homomorphism of topological O K -algebras O K [[t]][s π p ]/(π p s π p − t) −→ O K [[t]][s π ]/(πs π − t) that maps s π p → π −p ((1 + πs π ) p − 1), Then since and t → (1 + t) p − 1. π −p ((1 + πs π ) p − 1) − (s pπ − s π ) ∈ m K [[s π ]] π −p (η(s c 2 ) − 1) − s i c 0 2 ∈ m K [s c 2 ] + m K [[t]] [cf. Lemma 1.1; Proposition 1.6, (i)], it holds that • the morphism (U c 2 ) s → (U π p ) s corresponds to the homomorphism k[s π p ] −→ k[s c 2 ] over k that maps s π p → s i c 0 2 , while • the morphism (U π ) s → (U π p ) s corresponds to the homomorphism k[s π p ] −→ k[s π ] over k that maps s π p → s pπ − s π . 24 Thus, we conclude that the ﬁrst projection morphism Y s = (U c 2 ) s × (U πp ) s (U π ) s → (U c 2 ) s = Spec k[s c 2 ] corresponds to the natural injective homomorphism k[s c 2 ] → k[s c 2 , y]/(y p − y − s i c 0 2 ) over k. This completes the proof of assertion (ii), hence of Proposition 1.6. Remark 1.6.1. In the notation of Proposition 1.6, we observe that the ﬁrst projection morphism Y → U c 2 ﬁts into a commutative diagram Y −−−−→ U π θ Y ⏐ ⏐ ⏐ ⏐ f Y f ˜ | Uπ −−−−→ Spec O K [[t]] −−−−→ G m ι φ π | U π ⏐ ⏐ ⏐ ⏐ p f U c 2 −−−−→ U π p −−−−−→ Spec O K [[t]] −−−−→ G m , θ g φ π p | U π p ι where the ﬁrst vertical arrow f Y denotes the ﬁrst projection morphism Y → U c 2 ; the left-hand upper horizontal arrow θ Y denotes the second projection morphism Y → U π ; the vertical arrows are ﬁnite ﬂat morphisms of degree p [cf. Proposition 1.5, Remark 1.5.1]; the ﬁrst and third squares are cartesian; the second square is cartesian, up to taking the normalization of the ﬁber product that would make the second square “truly cartesian”. Remark 1.6.2. Let k be a ﬁeld of characteristic p; n a positive integer. Write C for the Artin-Schreier curve over k deﬁned by the equation y p − y = x n , where x and y are indeterminates; g C for the genus of C. Then it follows immediately from Hurwitz’s formula that g C = (n − 1)(p − 1) , 2 where n denotes the greatest positive integer that divides n and is prime to p. In particular, if n is not a power of p, then g C ≥ 1. [Indeed, by considering the Frobenius morphism, one reduces immediately to the case where n = n . Moreover, the computation of g C is immediate when n = 1 in light of the form of the equation y p − y = x. Thus, the computation of g C reduces to the computation of the genus of a tamely ramiﬁed cyclic covering of the projective line of degree n whose ramiﬁcation consists solely of p+1 totally ramiﬁed points.] 25 2 Resolution of nonsingularities for arbitrary hy- perbolic curves over p-adic local ﬁelds Let p be a prime number. In the present section, we apply certain con- structions involving p-divisible groups to extend the Artin-Schreier coverings constructed locally in §1 to coverings of an arbitrary hyperbolic curve. As a consequence, we prove that arbitrary hyperbolic curves over p-adic local ﬁelds satisfy RNS, i.e., “resolution of nonsingularities” [cf. Deﬁnition 2.2, (vii); The- orem 2.17]. This result may be regarded as a generalization of results obtained by A. Tamagawa and E. Lepage [cf. [Tama2], Theorem 0.2; [Lpg1], Theorem 2.7]. Historically [cf., e.g., the discussion in the Introduction to [Tama2]], the roots of these results of Tamagawa and Lepage may be traced back to the tech- nique of “passing to a covering with singular reduction of a given curve with smooth reduction over a p-adic local ﬁeld” applied in the proof of [PrfGC], The- orem 9.2 [cf. also Proposition 2.3, (xii), below]. Moreover, the techniques of [AbsTopII], §2, may be regarded as a sort of weak, pro-p version of Tamagawa’s RNS [cf. [AbsTopII], Remark 2.6.1]. In fact, the approach of the present sec- tion may be regarded as a sort of amalgamation of the techniques of [Lpg1] with the techniques of [AbsTopII], §2. At any rate, from a historical point of view, it is interesting to observe how various RNS results have been motivated by and indeed are deeply intertwined with various results in anabelian geometry [cf. Corollary 2.5, as well as Theorems 3.12, 3.13 in §3 below]. First, we begin by ﬁxing our conventions concerning models of hyperbolic curves [cf. [DM], Deﬁnition 1.1; [Knud], Deﬁnition 1.1]. Deﬁnition 2.1. Let K be a valuation ﬁeld; X a hyperbolic curve over K; X a scheme over O K . Then: (i) We shall say that X is a compactiﬁed model of X over O K if X is a proper, ﬂat, normal scheme of ﬁnite presentation over O K whose generic ﬁber is the [uniquely determined, up to unique isomorphism] smooth compactiﬁ- cation of X over K. (ii) Suppose that the cusps of X are K-rational. Then we shall say that X is a compactiﬁed semistable model of X over O K if X is a compactiﬁed model of X over O K such that the following conditions hold: • the geometric special ﬁber of X is a semistable curve [i.e., a reduced, connected curve each of whose nonsmooth points is an ordinary dou- ble point]; • the images of the sections Spec O K → X determined by the cusps of X [which we shall refer to as cusps of X ] lie in the smooth locus of X and do not intersect each other. Suppose that X is a compactiﬁed semistable model of X over O K . Then we shall say that X has split reduction if X s is split [i.e., each of the irreducible components and nodes of X s is geometrically irreducible]. 26 (iii) Suppose that X is a compactiﬁed semistable model of X over O K that has split reduction. Let L be a ﬁnite extension of K equipped with a valuation that extends the valuation on K; X ∗ a compactiﬁed semistable model of X L over O L such that X ∗ has split reduction and dominates X . Then we shall say that X ∗ is a toral compactiﬁed semistable model relative to X if each irreducible component of X s ∗ that maps to a closed point of X s [via the uniquely determined morphism X ∗ → X ] is normal of genus 0 and has precisely 2 nodes. Suppose that X ∗ is a toral compactiﬁed semistable model relative to X . Let v be a vertex of the dual graph associated to X s ∗ . Then we shall say that v is a toral semistable vertex of X ∗ if v corresponds to an irreducible component of X s ∗ that maps to a closed point of X . (iv) We shall say that X is a compactiﬁed stable model of X over O K if X is a compactiﬁed semistable model of X over O K such that X , together with the cusps of X , determines a pointed stable curve. (v) We shall say that X is a semistable model of X over O K if X is obtained by removing the cusps from a [uniquely determined, by Zariski’s Main Theorem, up to unique isomorphism] compactiﬁed semistable model of X over O K . Suppose that X is a semistable model of X over O K . Then we shall say that X has split reduction if X s is split [i.e., each of the irreducible components and nodes of X s is geometrically irreducible]. (vi) We shall say that X is a stable model of X over O K if X is obtained by removing the cusps from a [uniquely determined, by Zariski’s Main Theorem, up to unique isomorphism] compactiﬁed stable model of X over O K . (vii) We shall say that X has stable reduction over K if there exists a [necessarily unique, up to unique isomorphism] stable model of X over O K . Suppose that X is a stable model of X over O K . Then we shall say that X has split stable reduction over K if X s is split [i.e., each of the irreducible components and nodes of X s is geometrically irreducible]. Remark 2.1.1. It follows from elementary commutative algebra/scheme theory [cf. [EGAIV 2 ], Corollaire 6.1.2; [EGAIV 3 ], Proposition 12.1.1.5] that any com- pactiﬁed model as in Deﬁnition 2.1, (i), is of dimension 2 whenever K is a complete discrete valuation ﬁeld. Remark 2.1.2. In the notation of Deﬁnition 2.1, suppose that X is a proper hyperbolic curve over K. Then it follows immediately from the various deﬁni- tions involved that the notion of a compactiﬁed semistable model of X over O K coincides with the notion of a semistable model of X over O K . 27 Remark 2.1.3. In the notation of Deﬁnition 2.1, suppose that K is a complete discrete valuation ﬁeld, and that X has split stable reduction over K. Let X be a compactiﬁed semistable model of X over O K that has split reduction; φ : Y → X a morphism of compactiﬁed semistable models over O K that re- stricts to a connected ﬁnite étale covering Y → X over K. Then one veriﬁes immediately — by considering the map induced on irreducible components of the respective special ﬁbers by the necessarily ﬁnite, hence surjective morphism [induced by Y → X ] between the [two-dimensional, normal, integral] spectra of the completions of the local rings of X , Y at the closed points under consider- ation — that φ always maps a smooth closed point of Y that is isolated in the ﬁber of φ to a smooth closed point of X . Remark 2.1.4. In the notation of Remark 2.1.3, let X ∗ be a toral compactiﬁed semistable model relative to X ; e an edge of the dual graph associated to X s ; b a branch of the node e; v a toral semistable vertex of X ∗ that maps to [i.e., for which the corresponding irreducible component maps to the node corresponding e of the local ring of X at e is isomorphic to to] e. Recall that the completion O O K [[x, y]]/(xy − a), where a ∈ m K \ {0}, and x, y denote indeterminates chosen so that the ideal (x) (⊆ O K [[x, y]]/(xy − a)) corresponds to b. Observe that this ideal (x) is independent of the choice of x, y [cf. [Hur], §3.7, Lemma]. In particular, we obtain a homomorphism of local rings v ψ : O K [[x, y]]/(xy − a) −→ O v denotes the completion of the local ring of X ∗ at the generic point where O of the irreducible component of X ∗ corresponding to v. Write ord v (−) for the v whose normalization is determined by normalized valuation associated to O the condition that ord v (p) = 1 ∈ R. Thus, we obtain a rational number def 0 < ρ b,v = ord v (ψ(x)) < 1 ord v (ψ(a)) associated to b and v, which is in fact independent of the normalization of “ord v (−)”. If, moreover, we write b for the other branch of e, then one veriﬁes immediately that ρ b,v + ρ b ,v = 1. Finally, we observe that given any rational number ρ such that 0 < ρ < 1, there exist, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, an X ∗ and v as above such that ρ b,v = ρ. Indeed, it follows immediately from the theory of pointed stable curves, as ex- posed in [Knud], that, by possibly replacing K by a suitable ﬁnite extension ﬁeld of K and X by a suitable dense open subscheme of X, we may assume that X is the [unique, up to unique isomorphism] compactiﬁed stable model of X over O K , and that ρ may be written as a fraction whose denominator divides the positive integer v p (a). Then it follows again from the theory of pointed stable curves, as exposed in [Knud], that, if we take 28 • X ∗ to be the [unique, up to unique isomorphism] compactiﬁed stable model over O K of the hyperbolic curve obtained by removing from X a suitable K-rational point of X with center at e [cf. the construction of the displayed ∼ r X † ,x →) homomorphism “( O O K [[s, t]]/(st − π K ) → O K ” in the portion of the proof of Proposition 2.3, (iii), below, concerning the case where “x is a nonsmooth closed point of X † ”] and • v to be the unique irreducible component of X s ∗ that maps to a closed point of X s via the natural morphism X ∗ → X [cf. the morphism “X † [x] → X † ” in the portion of the proof of Proposition 2.3, (iii), below, concerning the case where “x is a nonsmooth closed point of X † ”], then ρ b,v = ρ, as desired. Remark 2.1.5. In the notation of Remark 2.1.4, we observe that, for a ﬁxed choice of b, the assignment v → ρ b,v ∈ Q that assigns to a toral semistable vertex v of X ∗ that maps to e the rational number ρ b,v is injective. Indeed, it follows immediately from the theory of pointed stable curves, as ex- posed in [Knud] — i.e., by adding ﬁnitely many suitably positioned cusps and then considering the various contraction morphisms that arise from eliminating cusps — that, to verify the asserted injectivity, it suﬃces to show that if v and w are the unique toral semistable vertices of X ∗ that map to e, which implies that there exists an edge e ∗ of the dual graph associated to X s ∗ that abuts to v and w, then ρ b,v = ρ b,w . To this end, we recall that X and X ∗ admit natural log structures [determined by the respective multiplicative monoids of regular functions invertible outside the respective special ﬁbers X s , X s ∗ ] such that the morphism X ∗ → X extends uniquely to a morphism of log schemes [cf., e.g., the subsection in Notations and Conventions entitled “Log schemes”; the discussion e ∗ of the local ring of of [Hur], §3.7, §3.8, §3.10]. Moreover, the completion O ∗ ∗ ∗ ∗ ∗ ∗ ∗ X at e is isomorphic to O K [[x , y ]]/(x y − a ), where a ∗ ∈ O K , and x ∗ , y ∗ denote indeterminates which may be chosen in such a way that the homo- e → O e ∗ induced by X ∗ → X maps morphism of topological O K -algebras O × ∗ N ∗ M x → (x ) · (π ) · u, for some unit u ∈ O e ∗ , some uniformizer π ∗ ∈ O K , some positive integer N , and some nonnegative integer M . [Indeed, the fact that N is necessarily positive follows immediately, in light of our assumptions on v and w, from well-known considerations in intersection theory on the Q-factorial normal schemes X ∗ and X .] Then the desired inequality ρ b,v = ρ b,w follows immediately from the fact that, up to a possible permutation of the labels “v” and “w”, it holds that x ∗ is invertible at the generic point of [the irreducible component corresponding to] v, but non-invertible at the generic point of [the irreducible component corresponding to] w. 29 Remark 2.1.6. In the notation of Remark 2.1.3, suppose further that Y has split reduction, and that the morphism Y → X is an isomorphism. Let X ∗ be a toral compactiﬁed semistable model relative to X , Y → X ∗ a morphism over X . Then observe that it follows immediately from Remark 2.1.3, together with Zariski’s Main Theorem, that each normal irreducible component of Y s that • maps to a closed point of X s , • is of genus 0, and • has precisely 1 node maps to a closed point of X s ∗ . In particular, it follows immediately from an iterated application of the above observation, together with the theory of pointed stable curves, as exposed in [Knud] — i.e., by adding ﬁnitely many suitably positioned cusps and then considering the various contraction morphisms that arise from eliminating cusps — that there exists a unique, up to unique isomorphism, toral compactiﬁed semistable model Y ∗ relative to X , together with a uniquely deter- mined morphism Y → Y ∗ of compactiﬁed semistable models over X , such that the following universal property is satisﬁed: if X † is a toral compactiﬁed semistable model relative to X such that the morphism Y → X admits a factorization Y → X † → X , then the morphism Y → X † admits a unique factorization Y → Y ∗ → X † . That is to say, Y ∗ may be thought of as a sort of “universal toralization [over X ]” of Y. In particular, it follows immediately from the existence of universal toralizations, together with the theory of pointed stable curves, as exposed in [Knud] — i.e., by adding ﬁnitely many suitably positioned cusps and then con- sidering the various contraction morphisms that arise from eliminating cusps — that the toral compactiﬁed semistable models relative to X form a directed inverse system. Deﬁnition 2.2. Let Σ ⊆ Primes be a nonempty subset; K a mixed character- istic complete discrete valuation ﬁeld of residue characteristic p; X a hyperbolic curve over K. Write Ω for the p-adic completion of [some ﬁxed] K. Then: (i) Let v be a valuation on a ﬁeld F that contains K. Write O v for the ring of integers determined by v; m v ⊆ O v for the maximal ideal of O v . Then we shall say that v is a p-valuation [over K] if O K = O v ∩ K [which implies that p ∈ m v ]. Here, the phrase “over K” will be omitted in situations where the base ﬁeld K is ﬁxed throughout the discussion. We shall say that v is primitive if it is a p-valuation such that the only prime ideal of O v that contains p is m v . We shall say that v is residue-transcendental def if it is a p-valuation whose residue ﬁeld k v = O v /m v is a transcendental extension of the residue ﬁeld of K. 30 is a universal (ii) In the situation of (i), suppose that v is a p-valuation, that X geometrically pro-Σ covering of X, and that F is a subﬁeld [in a fashion Ω ) compatible with the given inclusion K → F ] of the function ﬁeld K( X of X Ω . Then we shall say that v is point-theoretic if it arises from some point x̃ ∈ X(Ω), i.e., if O v ⊆ F is equal to the subring of F consisting Ω that are regular of elements ∈ F that determine rational functions on X at x̃, and whose value at x̃ is contained in O Ω ⊆ Ω. Thus, every point x̃ ∈ X(Ω) determines a corresponding point-theoretic valuation of F . (iii) In the situation of (ii), let Z → X be a connected ﬁnite étale covering → Z → X. Let Z be a compactiﬁed equipped with a factorization X semistable model of Z with split reduction. Then we shall write VE(Z) for the ﬁnite set [equipped with the discrete topology] of vertices and edges of the dual graph associated to Z s . Note that there is a notion of specialization/generization among elements of VE(Z), i.e., we shall say that • a vertex specializes to a node, or, alternatively, that a node generizes to a vertex, if the node abuts to the vertex; • a vertex specializes/generizes to a vertex if the two vertices coincide; • an edge specializes/generizes to an edge if the two edges coincide. For c 1 , c 2 ∈ VE(Z), if c 1 specializes to c 2 , or, equivalently, c 2 generizes to c 1 , then we shall write c 1 c 2 . By allowing Z and Z to vary, we thus obtain a topological space = lim VE(Z), VE( X) ←− def Z where the transition maps in the inverse limit are induced by the corre- sponding scheme-theoretic morphisms of compactiﬁed semistable models [which form a directed inverse system — cf. Proposition 2.3, (iii), below], that is to say, by mapping a vertex [i.e., irreducible component] or edge [i.e., node] to the smallest vertex [i.e., irreducible component] or edge [i.e., node] that contains its scheme-theoretic image. [Here, we recall that any such morphism of compactiﬁed semistable models always maps a smooth closed point that is isolated in the ﬁber of the morphism to a smooth closed as a VE- point — cf. Remark 2.1.3.] We shall refer to an element of VE( X) chain of X. Note that the notion of specialization/generization among el- ements of each VE(Z) determines [i.e., by considering each constituent set in the above inverse limit] a notion of specialization/generization among We shall say that an element c ∈ VE( X) is primitive elements of VE( X). if every generization of c is equal to c. 31 (iv) In the situation of (iii), let z c 1 , z c 2 ∈ VE(Z). Then we shall write δ(z c 1 , z c 2 ) ∈ Z for the integer δ such that the set of vertices contained in a path of minimal length between z c 1 and z c 2 on the dual graph of Z s is of cardinality δ + 1. Then we shall write Let c 1 , c 2 ∈ VE( X). def δ(c 1 , c 2 ) = sup δ(w c 1 , w c 2 ) ∈ Z ∪ {+∞}, Z where Z ranges over the set of compactiﬁed semistable models with split reduction of connected ﬁnite étale coverings Z → X equipped with a → Z → X; w c ∈ VE(Z) denotes the element determined factorization X i by c i for each i = 1, 2. (v) In the situation of (ii), suppose further that F contains the function ﬁeld of X. Then observe that v determines, by considering the centers K( X) associated to v on the various “Z” in the discussion of (iii), an element which we shall refer to as the center-chain associated to v. ∈ VE( X), In particular, any point x̃ ∈ X(Ω) determines, by considering the point- which we shall theoretic valuation associated to x̃, an element ∈ VE( X), refer to as the [point-theoretic] center-chain associated to x̃. Write x ∈ X(Ω) for the image of x̃ in X(Ω). Thus, the Gal( X/X K )-orbit of x̃ is completely determined by x. We shall refer to the Gal( X/X K )-orbit of the center-chain associated to x̃ as the [point-theoretic] orbit-center-chain associated to x [cf. the discussion of Remark 2.2.4 below]. (vi) In the situation of (ii), let Z → X be a connected ﬁnite étale covering → Z → X. Let Z be a compactiﬁed equipped with a factorization X semistable model with split reduction of Z. In the remainder of the dis- cussion of the present item (vi), all toral compactiﬁed semistable models relative to Z will be assumed to have generic ﬁbers that are equipped → Z. Write with the structure of a subcovering of the pro-covering X V(Z) (respectively, E(Z)) for the set of vertices (respectively, edges) of Z s . Thus, VE(Z) = V(Z) E(Z). For each c ∈ VE(Z), write V c for the set of equivalence classes of the set of vertices of toral compactiﬁed semistable models relative to Z that map to the closed subscheme of Z s corresponding to c, where we apply the equivalence relation induced by the dominant morphisms over Z of toral compactiﬁed semistable models relative to Z. Let e ∈ E(Z). Then we shall write V e for the union of V e and the vertices of V(Z) that abut to e. Observe that, for each toral compactiﬁed semistable model Z † relative to Z and 32 each c † ∈ VE(Z † ) that maps to the closed subscheme of Z s corresponding to c, the set V c † may be regarded, in a natural way [i.e., by considering the maps induced by dominant morphisms over Z of toral compactiﬁed semistable models relative to Z], as a subset of V c . We shall refer to such a subset V c † ⊆ V c as a basic open subset of V c . Thus, from the point of view of the natural bijection, determined by selecting a branch b of the edge e, between V e and the set of rational numbers ρ such that 0 < ρ < 1 [cf. Remarks 2.1.4, 2.1.5, 2.1.6], the basic open subsets ⊆ V e correspond precisely to the open intervals with rational endpoints of V e . In particular, it is natural to regard V c as being equipped with the topology determined by the open basis consisting of the basic open subsets ⊆ V c . We shall refer to as a quasi-basic open subset of V e any open subset of V e which is a union of a countable collection of basic open subsets ⊆ V e for which the relation of inclusion determines a total ordering. We shall refer to as a Dedekind cut of V e an unordered pair {D 1 , D 2 } of disjoint nonempty quasi-basic open subsets D 1 , D 2 ⊆ V e such that V e = D 1 ∪ D 2 . Write D e for the set of Dedekind cuts of V e . Note that the topology of the V w , where w ∈ V e \ V e , induces, in a natural way, a topology on the set def T e = V e D e [i.e., by taking as an open basis for the topology for T e the subsets of T e obtained as the intersections with T e of unions of an open subset U ⊆ V w , where w ∈ V e \ V e , with the set of Dedekind cuts {D 1 , D 2 } ∈ D e such that both D 1 and D 2 intersect U ]. Thus, def T e = V e D e . [with the induced topology] is homeomorphic to the open interval (0, 1) ⊆ R of the real line [cf. Remarks 2.1.4, 2.1.5, 2.1.6]. Write def VE(Z) tor = V(Z) ∪ T e . e∈E(Z) Thus, the discrete topology on V(Z), together with the topologies deﬁned above on the T e , determine a topology on VE(Z) tor . Moreover, there exists a noncontinuous [cf. Remark 2.2.1 below] natural surjective map Z : VE(Z) tor −→ VE(Z) that maps each T e to e. Finally, by allowing Z and Z to vary, we thus obtain a topological space tor = lim VE(Z) tor VE( X) ←− def Z [cf. the discussion of (iii)], together with a natural [not necessarily surjec- tive!] map tor −→ VE( X). X : VE( X) 33 (vii) We shall say that X satisﬁes Σ-RNS [i.e., “Σ-resolution of nonsingulari- ties” — cf. [Lpg1], Deﬁnition 2.1] if the following condition holds: Let v be a discrete residue-transcendental p-valuation on the function ﬁeld K(X) of X. Then there exists a connected geo- metrically pro-Σ ﬁnite étale Galois covering Y → X such that Y has stable reduction [over its base ﬁeld], and v coincides with the restriction [to K(X)] of a discrete valuation on the function ﬁeld K(Y ) of Y that arises from an irreducible component of the special ﬁber of the stable model of Y . Remark 2.2.1. In the notation of Deﬁnition 2.2, (vi), the natural surjective map Z is not continuous in general. Indeed, to see this, it suﬃces to observe that the inverse image of the closed subset consisting of a single edge is an open subset of VE tor (Z) that is not closed. Finally, we observe that one may also conclude from this noncontinuity of Z that X is not continuous. Remark 2.2.2. In the notation of Deﬁnition 2.2, (vii), suppose that Σ \ {p} is nonempty, and that X satisﬁes Σ-RNS. Then, by considering a suitable admis- sible covering of the stable model of “Y ” as in Deﬁnition 2.2, (vii), one veriﬁes immediately that one may assume that the normalization of the irreducible component that appears in Deﬁnition 2.2, (vii), is of genus ≥ 2. Remark 2.2.3. In the notation of Deﬁnition 2.2, (vii), we make the following observations. (i) Let L ⊆ Ω be a topological subﬁeld containing K that arises as the [topo- logical] ﬁeld of fractions of a mixed characteristic complete discrete valu- ation ﬁeld of residue characteristic p. Then let us observe that any compactiﬁed semistable model of X L over O L arises, after possibly replacing K and L, respectively, by suitable ﬁnite ex- tension ﬁelds of K and L, as the result of base-changing, from O K to O L , some compactiﬁed semistable model of X K over O K . Indeed, since every element of L admits arbitrarily close p-adic approxi- mations by elements of ﬁnite extension ﬁelds of K contained in K, this observation follows immediately by noting that it follows immediately from the well-known theory of pointed stable curves, as exposed in [Knud], that, after possibly replacing K and L, respectively, by suitable ﬁnite extension ﬁelds of K and L and possibly replacing X by some dense open subscheme of X, we may assume without loss of generality that the given compactiﬁed semistable model of X L over O L is in fact the [unique, up to unique iso- morphism] compactiﬁed stable model of X L , i.e., which necessarily arises as the result of base-changing, from O K to O L , the [unique, up to unique isomorphism] compactiﬁed stable model of X K over O K . 34 (ii) We maintain the notation of (i). Then let us observe that X satisﬁes Σ-RNS if and only if X L satisﬁes Σ-RNS. Indeed, this observation follows immediately, in light of the observation of (i), from Proposition 2.3, (ii), (iii), below; Proposition 2.4, (iv), below [cf. also Deﬁnition 2.2, (vii), as well as the discussion of the ﬁnal portion of the subsection in Notations and Conventions entitled “Fundamental groups”]. (iii) Next, let L be a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p that contains K as a topological subﬁeld. Then observe that it follows immediately from the well-known elementary theory of complete discrete valuation ﬁelds that L is isomorphic, as a topological K-algebra, to a ﬁeld “L” of the sort discussed in (i), (ii) if and only if [the valuation on] L is not residue-transcendental [relative to K], i.e., if and only if the residue ﬁeld of L is an algebraic extension of the residue ﬁeld of K. (iv) We maintain the notation of (iii). Then let us observe that if X L satisﬁes Σ-RNS, then the residue ﬁeld of L is an algebraic extension of the residue ﬁeld of K. Indeed, it suﬃces to verify this observation after replacing K by a ﬁnite extension ﬁeld of K. In particular, we may assume without loss of gener- ality [cf. Proposition 2.3, (iii)] that there exists a compactiﬁed semistable model X of X over O K . Then it follows immediately from the unique- ness, up to unique isomorphism, of compactiﬁed stable models [cf. also Deﬁnition 2.2, (vii), as well as the discussion of the ﬁnal portion of the subsection in Notations and Conventions entitled “Fundamental groups”; the stable reduction theorem of [DM], [Knud]], that if X L satisﬁes Σ-RNS, then any closed point of (X ⊗ O K O L ) s that arises as the center of a dis- crete residue-transcendental p-valuation on the function ﬁeld of X L is necessarily deﬁned over some algebraic extension of the residue ﬁeld of K. On the other hand, this contradicts the existence of discrete residue- transcendental p-valuations on the function ﬁeld of X L that arise as the local rings of generic points of exceptional divisors of blow-ups of smooth closed points of (X ⊗ O K O L ) s that are not deﬁned over some algebraic extension of the residue ﬁeld of K. This completes the proof of the above observation. (v) We maintain the notation of (iii). Then we observe further that, under the assumption that X satisﬁes Σ-RNS, it holds that the residue ﬁeld of L is an algebraic extension of the residue ﬁeld of K if and only if X L satisﬁes Σ-RNS. 35 Indeed, necessity follows formally from the observations of (ii), (iii), while suﬃciency follows formally from the observation of (iv). Remark 2.2.4. In the context of Deﬁnition 2.2, we recall from the general theory of valuations the following well-known basic facts. Let L be a ﬁeld equipped with a valuation v, M a ﬁnite normal extension ﬁeld of L. Write O v for the ring of integers of L with respect to v, m v ⊆ O v for the maximal ideal of O v , O M for the integral closure of O v in M , v M/L for the set of valuations on M that extend v, and Aut(M/L) for the group of automorphisms of M that restrict to the identity on L. If w ∈ v M/L , then we shall write O w for the ring of integers of def M with respect to w, m w ⊆ O w for the maximal ideal of O w , p w = O M ∩ m w . Then the set v M/L is nonempty [cf. [EP], Theorem 3.1.1], and the natural action of Aut(M/L) on v M/L is transitive [cf. [EP], Theorem 3.2.14]. Moreover, O M = O w w∈v M/L [cf. [EP], Theorem 3.1.3, (2)]; the assignment v M/L w → p w determines a bijective correspondence between v M/L and the set of prime ideals of O M that lie over m v , and, for w ∈ v M/L , O w = (O M ) p w [cf. [EP], Theorem 3.1.1; [EP], Theorem 3.2.13]. In this situation, if we assume further that v is real, and that L is complete with respect to v, then v M/L is of cardinality 1 [cf. [Neu], Chapter II, Theorem 4.8]. [Here, we recall that if v is not real, then O v does not, in general, satisfy Hensel’s Lemma, i.e., even if L is complete with respect to v [cf. [EP], Remark 2.4.6].] More generally, if v is real, then L admits a [cf. [EP], Theorem 1.1.4], which is a henselian ﬁeld [cf. natural completion L [Neu], Chapter II, Theorem 4.8; the discussion preceding [EP], Lemma 4.1.1] and contains, up to natural isomorphism, the henselization L h of L [cf. [EP], the discussion preceding Theorem 5.2.2] as a subﬁeld, i.e., L h ⊆ L [cf. [EP], Corollary 4.1.5; [EP], Corollary 5.2.3; the discussion of Case 2 in the proof of [EP], Theorem 6.3.1]. The various basic properties stated in the following Proposition 2.3 consist of elementary results that are essentially well-known or implicit in the literature [cf. Remarks 2.3.2, 2.3.3 below], but we give [essentially] self-contained state- ments and proofs here in the language of the present discussion for the sake of completeness. Proposition 2.3 (Basic properties of models of hyperbolic curves). Let K be a mixed characteristic complete discrete valuation ﬁeld of residue charac- teristic p; X a hyperbolic curve over K. Write K(X) for the function ﬁeld of X. Then the following hold: 36 (i) Let R ⊆ K(X) be a ﬁnitely generated normal O K -subalgebra whose ﬁeld of fractions coincides with K(X). Then Spec R arises as an open subscheme of a compactiﬁed model of X over O K . (ii) Let v be a discrete residue-transcendental [cf. Remark 2.3.1 below] p- valuation on K(X). Then v arises as the discrete valuation associated to an irreducible component of the special ﬁber of a compactiﬁed model of X over O K . (iii) Let X be a compactiﬁed model of X over O K equipped with the action of a ﬁnite group G by O K -linear automorphisms [which thus restrict to K- linear automorphisms of X]. Then, after possibly replacing K by a ﬁnite ﬁeld extension of K, there exists a compactiﬁed semistable model of X over O K that dominates X and is stabilized by the action of G on X. (iv) Let Y → X be a [connected] ﬁnite étale Galois covering of hyperbolic curves over K, Y sst a compactiﬁed semistable model of Y over O K that def is stabilized by G = Gal(Y /X). Write X for the quotient of Y sst by the natural action of G on Y sst . Then X is a compactiﬁed semistable model of X over O K , and the images of smooth points of Y s sst via the natural morphism Y s sst → X s are smooth points of X s . Moreover, the image of a node of Y s sst via the natural morphism Y s sst → X s is a node of X s if and only if G does not permute the branches of the node. In particular, • the dual graph of X s may be reconstructed from the dual graph of Y s sst , together with the action of G on the dual graph of Y s sst . Finally, • this reconstruction procedure is functorial, with respect to maps of vertices/edges to vertices/edges [i.e., as in the discussion of Deﬁ- nition 2.2, (iii)], on the category of [connected] ﬁnite étale Galois coverings of X over K. (v) Suppose that we are in the situation of Deﬁnition 2.2, (ii), (iii), (iv), (v). to its associated Then the assignment that maps a p-valuation on K( X) center-chain determines bijections as follows: p-valuations on K( X) ∼ → VE( X), primitive p-valuations on K( X) ∼ prim , → VE( X) ∼ pt-th , → VE( X) X(Ω) ∼ pt-th /Gal( X/X X(Ω) → VE( X) K ), prim ⊆ VE( X) denotes the subset of primitive VE-chains, where VE( X) pt-th and VE( X) ⊆ VE( X) denotes the subset of point-theoretic center- chains. 37 Write R c ⊆ (vi) Suppose that we are in the situation of (v). Let c ∈ VE( X). K( X) for the valuation ring of the p-valuation associated to c [cf. (v)]. Then it holds that R c = lim O Z,z c , −→ Z where the direct limit ranges over the set of compactiﬁed semistable models with split reduction Z of the domain curves of connected ﬁnite étale cov- → Z → X; z c denotes the erings Z → X equipped with a factorization X center on Z determined by R c ; the transition maps in the direct limit are induced by the corresponding scheme-theoretic morphisms of compactiﬁed semistable models [cf. the discussion of Deﬁnition 2.2, (iii)]. (vii) Suppose that we are in the situation of (v). Suppose, moreover, that X is is primitive if and only if it is either proper. Then a p-valuation of K( X) real or point-theoretic. Equivalently, consists of the disjoint • the set of primitive p-valuations of K( X) and the union of the non-point-theoretic real p-valuations of K( X) point-theoretic p-valuations of K( X). an for the topological pro-Berkovich space as- In particular, if we write X sociated to [i.e., the inverse limit of the underlying topological spaces of then the Berkovich spaces associated to the ﬁnite subcoverings of ] X, may be naturally identiﬁed • the set of primitive p-valuations of K( X) an with the underlying set of X . (viii) Suppose that we are in the situation of (vii). Then there exists a natural commutative diagram of maps of sets an X ⏐ ⏐ ∼ tor −−−−→ VE( X) θ X ⏐ ⏐ X prim −−−−→ VE( X), VE( X) ι X where the upper horizontal arrow θ X is a homeomorphism [cf. Remark 2.3.3 below]; the lower horizontal arrow ι X denotes the natural inclu- sion; the left-hand vertical arrow denotes the bijection obtained by form- ing the composite of the natural identiﬁcation that appears in the state- ment of (vi) with the second bijection in the display of (v); the right- hand vertical arrow X denotes the natural morphism [cf. Deﬁnition 2.2, (vi)]. In particular, X is injective and in fact admits a natural split- → VE( X) tor [i.e., such that τ ◦ is the identity on ting τ X : VE( X) X X tor ]. On the other hand, neither ι nor is surjective [cf. Remark VE( X) X X 2.3.4 below]. 38 be distinct (ix) Suppose that we are in the situation of (v). Let c 1 , c 2 ∈ VE( X) elements. Then one of the following conditions holds: • δ(c 1 , c 2 ) = +∞. such • δ(c 1 , c 2 ) = 0, and there exists a unique element c 3 ∈ VE( X) that c 3 c 1 and c 3 c 2 . In particular, if c 1 and c 2 are distinct primitive elements, then it holds that δ(c 1 , c 2 ) = +∞. Then the (x) Suppose that we are in the situation of (v). Let c ∈ VE( X). cardinality of the set \ {c} | c c} {c ∈ VE( X) is at most 1. (xi) Suppose that we are in the situation of (v). Let Σ ⊆ Primes be a subset; l ∈ Σ \ {p}; H ⊆ G K a closed subgroup such that the restriction to H of the l-adic cyclotomic character of K has open image, and, moreover, the intersection H ∩ I K of H with the inertia subgroup I K of G K admits (Σ) a surjection to [the proﬁnite group] Z l ; s : H → Π X a section of the (Σ) restriction to H of the natural surjection Π X G K . Then there exists prim that is ﬁxed by the restriction, via s, to H of an element c ∈ VE( X) (Σ) prim ⊆ VE( X). In particular, if X is the natural action of Π X on VE( X) an an proper, then there exists an element c ∈ X [cf. (vii)] that is ﬁxed by (Σ) the restriction, via s, to H of the natural action of Π X on the topological an . pro-Berkovich space X (xii) Let Σ ⊆ Primes be a subset of cardinality ≥ 2 that contains p. Then there exists a connected geometrically pro-Σ ﬁnite étale Galois covering X † → X satisfying the following conditions: • X † has split stable reduction. • Write X † for the [unique, up to unique isomorphism] stable model of X † . Then X s † is singular, and every irreducible component of X s † is a smooth curve of genus ≥ 2. Proof. First, we verify assertion (i). Since R is a ﬁnitely generated algebra over O K (⊆ R), it follows that Spec R admits an embedding over O K into † N N -dimensional aﬃne space A N O K for some positive integer N . Write Z ⊆ P O K N N for the scheme-theoretic closure of the image of Spec R in P O K (⊇ A O K ); Z for the normalization of Z † . Thus, the structure sheaf O Z is p-torsion-free, hence ﬂat over O K . Since, moreover, Z † is [of ﬁnite type over the complete discrete valuation ring O K , hence] excellent, it follows that Z is a proper, ﬂat scheme of ﬁnite type over O K , whose generic ﬁber may be identiﬁed with the [uniquely 39 determined, up to unique isomorphism] smooth compactiﬁcation of X over K. This completes the proof of assertion (i). Next, we verify assertion (ii). Write A ⊆ K(X) for the discrete valuation ring associated to v. Note that A may be written as the direct limit [i.e., in fact, union] of a direct system of ﬁnitely generated subalgebras {A i ⊆ A} i∈I over O K . Moreover, since the ﬁeld extension K ⊆ K(X) is ﬁnitely generated, and each A i is [ﬁnitely generated over the complete discrete valuation ring O K , hence] excellent, we may assume without loss of generality, i.e., by replacing A i by its normalization in K(X), that each A i is normal with ﬁeld of fractions equal to K(X). Write p i ⊆ A i for the prime ideal determined by the maximal ideal of A. Thus, since v is a p-valuation [over K], it follows immediately that each of the natural inclusions O K → (A i ) p i → A is a homomorphism of local rings. Next, let us observe that since the residue ﬁeld extension determined by the natural inclusion O K ⊆ A of local rings is assumed to be transcendental, it follows that there exists an element i ∈ I such that the residue ﬁeld k(p i ) of p i is a transcendental extension of the residue ﬁeld of O K . Let Z be a compactiﬁed model of X over O K that contains Spec A i as an open subscheme [cf. Proposition 2.3, (i)]. Then since Z is of dimension 2 [cf. Remark 2.1.1], it follows that the height of p i is equal to 1 or 2. On the other hand, if p i is of height 2, then it follows that p i corresponds to a closed point of Z s , hence that k(p i ) is a ﬁnite extension of the residue ﬁeld of O K , i.e., in contradiction to our assumption of transcendality. Thus, we conclude that p i is of height 1, hence that (A i ) p i is a discrete valuation ring whose ﬁeld of fractions is equal to K(X). But this implies that (A i ) p i = A. This completes the proof of assertion (ii). Next, we verify assertion (iii). First, we observe that, after possibly replac- ing K by a suitable ﬁnite extension ﬁeld of K, there exists a G-equivariant ﬁnite morphism X → P 1 O K to the projective line over O K [equipped with the trivial action by G]. Indeed, since O K is a complete discrete valuation ring, by deforming any [suitably large positive power of a] very ample line bundle on the projective curve X s , we obtain a very ample line bundle L on X , hence, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, a pair of global sections σ 1 , σ 2 of the line bundle L such that the G-orbit of the zero locus of σ 1 is disjoint from the G-orbit of the zero locus of σ 2 . Thus, for i = 1, 2, the product σ i G of the G-translates of σ i determines a global section of the [still very ample!] tensor product L G of G-translates of L such that σ 1 G and σ 2 G still have disjoint zero loci, hence determine a G-equivariant ﬁnite morphism X → P 1 O K over O K , as desired. Fix such a ﬁnite morphism, and write f ∈ K(X) for the rational function on X determined by the standard coordinate function on P 1 O K . Next, we recall that it follows from the stable reduction theorem [cf. [DM], [Knud]] that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, we may assume without loss of generality that every closed point in the support Supp(f ) [in the smooth compactiﬁcation of X] of the principal divisor associ- ated to f is K-rational, and that X has stable reduction over K. Moreover, by replacing X by a suitable G-stable open subscheme of X, we may assume with- out loss of generality that Supp(f ) is contained in the set of cusps of X. Write X † for the compactiﬁed stable model of X over O K ; E ⊆ X † for the reduced 40 closed subscheme determined by the set of closed points where an irreducible component of the zero divisor of f on X † intersects an irreducible component of the divisor of poles of f on X † . Thus, the action of G on X extends to X † . X † ,x for the completion of the local ring of X † at x. Fix a Let x ∈ E. Write O uniformizer π K ∈ O K . Next, suppose that x is a smooth closed point of X † . Then there exist nonzero integers a, b of opposite sign and a unit u ∈ (O K [[t]]) × [where t denotes an indeterminate], together with an isomorphism of topological O K -algebras ∼ ∼ X † ,x ( → X † ,x → O K [[t]], such that the image of f in the ﬁeld of fractions of O O b O K [[t]]) is of the form u·t a ·π K . Next, observe that, by replacing K by a suitable ﬁnite extension ﬁeld of K [so it may no longer be the case that the element “π K ” is a uniformizer of O K !], we may assume without loss of generality that there −b exists an element γ ∈ O K such that γ a = π K . Write x η for the K-valued point of the smooth compactiﬁcation of X determined by the section of the structure morphism X † → Spec O K corresponding to the homomorphism of topological O K -algebras ∼ X † ,x →) ( O O K [[t]] −→ O K that maps t → γ ∈ O K ; x η for the K-valued point of the smooth compactiﬁca- tion of X determined by the section of the structure morphism X † → Spec O K corresponding to the homomorphism of topological O K -algebras ∼ X † ,x →) O K [[t]] −→ O K ( O that maps t → 0 ∈ O K ; X † [x] for the compactiﬁed stable model of X \ {x η , x η } over O K . Thus, it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that the natural inclusion X \ {x η , x η } → X determines a natural birational, dominant morphism X † [x] → X † . Finally, we observe that it follows immediately from the various deﬁnitions involved that the rational function f is a unit [at x η , hence] at the generic point of the unique irreducible component of (X † [x]) s that maps to a closed point of X s † ; in particular, the zero divisor of f does not intersect the divisor of poles of f in some Zariski neighborhood of this irreducible component. Next, suppose that x is a nonsmooth closed point of X † . Then since the X † ,x ) s are Q-Cartier divisors, it follows two irreducible components of (Spec O r that there exist positive integers a, b, r and a unit u ∈ (O K [[s, t]]/(st − π K )) × [where s, t denote indeterminates], together with an isomorphism of topological ∼ r X † ,x → O K -algebras O O K [[s, t]]/(st−π K ), such that the image of some positive ∼ r power of f in the ﬁeld of fractions of O X † ,x ( → O K [[s, t]]/(st − π K )) is of the a −b form u·s ·t . Next, observe that, by replacing K by a suitable ﬁnite extension ﬁeld of K [so it may no longer be the case that the element “π K ” is a uniformizer of O K !], we may assume without loss of generality that there exists an element γ ∈ O K such that γ a+b = π K . Write x η for the K-valued point of the smooth compactiﬁcation of X corresponding to the section of the structure morphism X † → Spec O K induced by the homomorphism of topological O K -algebras ∼ X † ,x →) O K [[s, t]]/(st − π r ) −→ O K ( O K 41 def that maps s → γ br ∈ O K , t → γ ar ∈ O K ; x η = x η ; X † [x] for the compactiﬁed stable model of X \ {x η , x η } = X \ {x η } over O K . Thus, it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that the natural inclusion X \ {x η , x η } = X \ {x η } → X determines a natural birational, dom- inant morphism X † [x] → X † . Finally, we observe that it follows immediately from the various deﬁnitions involved that the rational function f is a unit [at x η , hence] at the generic point of the unique irreducible component of (X † [x]) s that maps to a closed point of X s † ; in particular, the zero divisor of f does not in- tersect the divisor of poles of f in some Zariski neighborhood of this irreducible component. Next, observe that the underlying set of E is ﬁnite. Thus, by replacing X by a suitable G-stable open subscheme of X, we may assume without loss of generality that for each x ∈ E, the K-valued points x η , x η constructed above are contained in the set of cusps of X. Then it follows immediately from the above discussion that the zero divisor of f on X † does not intersect the divisor of poles of f on X † . But this implies that f determines a G-equivariant domi- nant morphism X † → P 1 O K over O K whose restriction to the respective generic ﬁbers coincides with the restriction to the respective generic ﬁbers of the ﬁnite morphism X → P 1 O K over O K constructed above. Thus, since X † is normal, we conclude that the morphism X † → P 1 O K admits a factorization X † → X → P 1 O K , as desired. This completes the proof of assertion (iii). Next, we verify assertion (iv). First, we observe that since the operation of forming the quotient of Y sst by G commutes with ﬂat base-change, one veriﬁes immediately that it suﬃces to verify assertion (iv) after performing any ﬁnite, faithfully ﬂat base-change from O K to the ring of integers in a ﬁnite extension ﬁeld of K. In particular, by replacing K by a suitable ﬁnite extension ﬁeld of K, we may assume without loss of generality that the cusps of X are K-rational, and that X has stable reduction over K [cf. [DM], [Knud]]. [In fact, these conditions are satisﬁed even if one does not pass to a ﬁnite extension ﬁeld of the original given K, but we omit a proof of this fact since it is not logically necessary for the present discussion.] Write X st for the compactiﬁed stable model of X over O K . Next, let us observe that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, one may regard Y sst as the compactiﬁed stable model associated to the hyperbolic curve Y obtained by removing from Y a collection of G-orbits of K-rational points of Y such that the cardinality of the set of G-orbits of closed points of Y s sst contained in the intersection of any irreducible component of Y s sst with the image of the corresponding collection of O K -rational points of Y sst is ≥ 3. In particular, one veriﬁes immediately that, by replacing Y by Y , we may assume without loss of generality that the cardinality of the set of closed points of each irreducible component of X s that lie in the image of the cusps of Y sst is ≥ 3. Next, let us observe that it follows immediately from the deﬁnition of the natural quotient morphism Y sst → X that the natural morphism Y sst → X st over O K induced by the morphism Y → X [cf. [ExtFam], Theorem A] admits 42 a factorization Y sst −→ X −→ X st , where we note that it follows immediately from the deﬁnition of X that X is a compactiﬁed model of X over O K . Thus, it follows immediately from the above discussion of Y sst , together with the theory of pointed stable curves, as exposed in [Knud], that the morphism X → X st is birational and quasi-ﬁnite, hence, by Zariski’s Main Theorem, an isomorphism. In particular, we conclude that X is a compactiﬁed semistable model of X over O K . Moreover, it follows immediately from Remark 2.1.3 that the natural morphism Y s sst → X s maps smooth points of Y s sst to smooth points of X s . Next, let e Y ∈ Y s sst be a node. Write e X ∈ X s for the image of e Y via the natural morphism Y s sst → X s ; G e Y ⊆ G for the stabilizer of e Y in G; Y e sst for the Y spectrum of the completion of the local ring of Y sst at e Y ; X e X for the spectrum of the completion of the local ring of X at e X . Thus, G e Y acts naturally on Y e sst ; X e X may be identiﬁed with the quotient of Y e sst by the action of G e Y ; the Y Y set B Y of irrreducible components of (Y e sst ) may be identiﬁed with the set [of Y s cardinality 2] of branches of e Y ; the set B X of irrreducible components of (X e X ) s is of cardinality 1 if and only if e X is a smooth point of X s and may be identiﬁed with the set [of cardinality 2] of branches of e X whenever e X is a node. On the other hand, it follows immediately from elementary commutative algebra that the set B X may be naturally identiﬁed with the set of G e Y -orbits of B Y . The remaining portion of assertion (iv) now follows formally. This completes the proof of assertion (iv). Write Next, we verify assertions (v) and (vi). Let c ∈ VE( X). def R c = lim O Z,z c , −→ Z where • Z ranges over the compactiﬁed semistable models with split reduction of the domain curves of connected ﬁnite étale coverings Z → X equipped → Z → X; with a factorization X • c Z denotes the irreducible component or node of Z s determined by c; • z c denotes the generic point of the intersection of the [closed irreducible] images in Z s of the c Z † associated to compactiﬁed semistable models with split reduction of domain curves of connected ﬁnite étale coverings → Z † → Z → X such that the Z † → X equipped with a factorization X † morphism Z → Z extends to a morphism Z † → Z; • the transition maps in the direct limit are the homomorphisms of local rings induced by the corresponding scheme-theoretic morphisms of com- pactiﬁed semistable models [which form a directed inverse system — cf. Proposition 2.3, (iii)]. 43 Then it follows immediately from the various deﬁnitions involved that the ﬁeld that O K ⊆ R c , and that R c is a local of fractions of R c coincides with K( X), c ⊆ K( X) be a valuation domain whose maximal ideal m R c contains p. Let R c for the ring that dominates R c [cf., e.g., [EP], Theorem 3.1.1]. Write m R c ⊆ R c . Thus, since O K ⊆ O ⊆ R c ⊆ R c and p ∈ m R ⊆ m , we maximal ideal of R c K R c c , i.e., that the valuation determined by the valuation conclude that O K = K ∩ R c is a p-valuation. ring R c may be written as the direct limit [i.e., in fact, union] of a direct Note that R c } i∈I over O K . Moreover, system of ﬁnitely generated subalgebras {R i ⊆ R since R i is [ﬁnitely generated over the complete discrete valuation ring O K , hence] excellent, by replacing R i by its normalization in the subﬁeld of K( X) generated by the ﬁeld of fractions of R i and some suitable ﬁnite extension ﬁeld of K, we may assume without loss of generality that R i is normal, and that there exists a compactiﬁed semistable model with split reduction Z i of the domain curve of a connected ﬁnite étale covering Z i → X equipped with a → Z i → X such that Spec R i arises as an open subscheme of factorization X def Z i [cf. Proposition 2.3, (i), (iii)]. Write p i = m R c ∩ R i ⊆ R i . Thus, if we write z i for the point of Z i that corresponds to p i , then it holds that c = lim O Z ,z . R i i −→ i∈I On the other hand, observe that, since m R c ∩ R c = m R c , it follows immediately from the various deﬁnitions involved that each “O Z i ,z i (= (R i ) p i )” of the above direct limit appears as one of the “O Z,z c ” in the direct limit used to deﬁne R c . c , hence that R c = R c , i.e., that R c c ⊆ R c ⊆ R In particular, we conclude that R is the valuation ring associated to a p-valuation. Thus, in summary, we obtain a natural map −→ p-valuations on K( X) VE( X) c → R c . Moreover, one veriﬁes immediately that this map that maps VE( X) deﬁnes an inverse to the natural map p-valuations on K( X) −→ VE( X) in the statement of assertion (v), hence that both of these maps are bijective. Since both of these maps are manifestly compatible with specialization/generization, we thus conclude that these induce a bijection primitive p-valuations on K( X) ∼ prim → VE( X) as in the statement of assertion (v). Thus, to complete the proof of assertion (v), it suﬃces to verify that the natural map X(Ω) −→ p-valuations on K( X) 44 [i.e., that assigns to an element of X(Ω) the associated point-theoretic valuation on K( X)] is injective. To this end, let x̃ ∈ X(Ω). Write v for the point-theoretic valuation on K( X) associated to x̃. Then observe that x̃ is deﬁned over K if and is strict. If this inclusion is strict, then the only if the inclusion O v · K ⊆ K( X) is the valuation ring determined [i.e., in the usual sense subring O v · K ⊆ K( X) of the classical theory of one-dimensional function ﬁelds over algebraically closed is strict, the ﬁelds] by x̃. In particular, whenever the inclusion O v · K ⊆ K( X) point x̃ ∈ X(K) (⊆ X(Ω)) is completely determined by v. Thus, it remains to In this case, the valuation v is real, and consider the case where O v ·K = K( X). induces, by passing to the respective completions, an the inclusion K ⊆ K( X) with respect to v. In particular, isomorphism of Ω with the completion of K( X) we obtain a natural homomorphism K( X) → Ω, which completely determines the point x̃ ∈ X(Ω). This completes the proof of assertion (v). Assertion (vi) follows immediately from the proof of assertion (v). Next, we verify assertion (vii). Let x ∈ X(Ω). Write v for the point-theoretic valuation on K( X) associated to x; φ x : O v → O Ω for the homomorphism obtained by evaluating rational functions at x. Let q ⊆ O v be a prime ideal that contains p. Then observe that it follows immediately from the construction of v that p 1 ·Ker(φ x ) ⊆ Ker(φ x ). Since p ∈ q, we thus conclude that Ker(φ x ) ⊆ q. On the other hand, observe that [it follows immediately from the construction of v that] this inclusion implies that q contains [hence coincides with] the radical of the ideal (p, Ker(φ x )) ⊆ O v , which is easily seen to be equal to the maximal ideal m v of O v . Thus, we conclude that v is primitive. Let a ∈ m v . Then since v is real, Next, let v be a real p-valuation on K( X). there exists a positive integer N such that a N ∈ (p). In particular, any prime ideal that contains p contains [hence coincides with] m v . Thus, we conclude that v is primitive. For each z ∈ O v , write Next, let v be a primitive p-valuation on K( X). for the O v -subalgebra generated by 1 . Thus, if (K ⊆) (O v ) p = (O v ) z ⊆ K( X) z then it follows immediately from the classical theory of one-dimensional K( X), function ﬁelds over algebraically closed ﬁelds that v is a point-theoretic valuation. Note Therefore, we may assume without loss of generality that (O v ) p = K( X). that this implies that for each x ∈ O v \ {0}, there exist a positive integer N and y ∈ O v such that p N = xy. Moreover, in this situation, it holds that m K O v = m v . Indeed, since v is a p-valuation, the inclusion m K O v ⊆ m v is immediate. Now suppose that there exists an element x ∈ m v \ m K O v . Then it follows that 1 p ∈ (O v ) x , hence that there exists a prime ideal p v of O v such that x ∈ p v , and p ∈ p v . On the other hand, since v is primitive, we conclude that x ∈ m v = p v , a contradiction. This completes the proof of the equality in the above display. Note that this equality implies that for each x ∈ m v \ {0}, there exist a positive integer N and y ∈ O v such that x N = py. In particular, it follows immediately from the various deﬁnitions involved that v coincides with the real valuation 45 determined by the assignment O v \ {0} x → sup{ ∈ Q | x ∈ p · O v } = inf{ ∈ Q | x −1 ∈ p − · O v } ∈ R. This completes the proof of assertion (vii). Next, we verify assertion (viii). First, let us observe that it follows immedi- ately from the ﬁnal portion of Proposition 2.3, (vii), that the natural map an −→ VE( X) X the center-chain associated to the — i.e., that assigns to a valuation on K( X) valuation — admits a factorization ∼ an → prim −→ VE( X), X VE( X) ι X where the ﬁrst arrow is a bijection, and the second arrow ι X denotes the natural inclusion. On the other hand, it follows immediately from the discussion of the tor ” in Deﬁnition ratios “ρ b,v ” in Remark 2.1.4 and the construction of “VE( X) also admits a factorization an → VE( X) 2.2, (vi), that this natural map X tor −→ VE( X), an −→ VE( X) X θ X X where the ﬁrst map θ X is deﬁned by considering ratios “ρ b,v ” as in Remark 2.1.4 tor ” in Deﬁnition 2.2, (vi)], and the second [cf. also the construction of “VE( X) arrow X is the natural map discussed in the ﬁnal portion of Deﬁnition 2.2, (vi). In particular, we obtain a commutative diagram of maps of sets an X ⏐ ⏐ tor −−−−→ VE( X) θ X ⏐ ⏐ X prim −−−−→ VE( X). VE( X) ι X Note that the commutativity of the diagram already implies that θ X is injective. Moreover, one veriﬁes immediately — i.e., by considering suitable “v” as in Remark 2.1.4 — that each composite map θ Z tor −→ VE(Z) tor an −→ VE( X) X θ X — where the second arrow is the natural projection arising from the inverse limit tor [cf. Deﬁnition 2.2, (vi)] — has dense image. Thus, in the deﬁnition of VE( X) an [cf. [Brk], Theorem 1.2.1], together it follows from the compactness of X with the easily veriﬁed fact [cf. the construction of Deﬁnition 2.2, (vi)] that VE(Z) tor is Hausdorﬀ, that to verify that θ X is a homeomorphism, it suﬃces to verify that each map θ Z is continuous. Moreover, once one knows that θ X is a homeomorphism, one may construct a natural splitting τ X as in the statement 46 of Proposition 2.3, (viii), by constructing a natural splitting of the natural → VE( X) prim inclusion ι X . On the other hand, such a natural splitting VE( X) of ι X is implicit in the content of Proposition 2.2, (x) [which will be veriﬁed below, independently of the present assertion (viii)], i.e., one assigns to each the unique generization ∈ VE( X) prim of c. nonprimitive element c ∈ VE( X) Thus, in summary, to complete the proof of assertion (viii), it suﬃces to verify that each map an −→ VE(Z) tor θ Z : X as in the above discussion is continuous. Let Z † be a toral compactiﬁed semistable model relative to Z. Then we shall refer to an open subscheme U of Z s † as a componentwise open of Z † if U is an open subscheme of Z s † whose underlying open subset is the complement of a node or an irreducible component of Z s † . Observe that it follows immediately from the construction of VE(Z) tor given in Deﬁnition 2.2, (vi), that each componentwise open of each toral compactiﬁed semistable model relative to Z determines, in a natural way, a closed subset of VE(Z) tor . We shall refer to the closed subsets of VE(Z) tor obtained in this way as componentwise closed subsets of VE(Z) tor . Note that it follows immediately from the construction of VE(Z) tor given in Deﬁnition 2.2, (vi), that the com- plements of the componentwise closed subsets of VE(Z) tor form an open basis of the topology of VE(Z) tor . Thus, to complete the proof of the contininuity of θ Z , it suﬃces to verify that the inverse image via θ Z of any componentwise an . But this follows immediately from closed subset of VE(Z) tor is closed in X the deﬁnition of the topology of the Berkovich spaces [cf. the discussion of [Brk], an ” in the §1.1, §1.2] that appear in the inverse limit that is used to deﬁne “ X statement of Proposition 2.3, (vii). This completes the proof of assertion (viii). Next, we verify assertion (ix). In the following, we assume that δ(c 1 , c 2 ) = +∞. Let us ﬁrst consider the case where δ(c 1 , c 2 ) ≥ 1. Then it follows immediately from the deﬁnition of δ(−, −) that there exists a compactiﬁed semistable model Z with split reduction of a connected ﬁnite étale covering Z → X equipped → Z → X such that δ(z c , z c ) ≥ 1 [cf. the notation with a factorization X 1 2 of Proposition 2.3, (vi)]. In particular, there exists a node e of Z s that does not coincide with z c 1 or z c 2 , and whose corresponding edge lies on a path of minimal length between z c 1 and z c 2 . On the other hand, by considering suitable torally compactiﬁed semistable models relative to Z at e, we conclude that δ(c 1 , c 2 ) = +∞, in contradiction to our assumption that δ(c 1 , c 2 ) = +∞. Thus, to complete the proof of assertion (ix), it suﬃces to consider the case where δ(c 1 , c 2 ) = 0. Let us ﬁrst observe that the condition that δ(c 1 , c 2 ) = 0 such that c 3 c 1 and c 3 c 2 . implies the existence of an element c 3 ∈ VE( X) Finally, we verify the uniqueness of such an element c 3 ∈ VE( X). Let c 4 ∈ VE( X) be such that c 3 = c 4 , c 4 c 1 , and c 4 c 2 . Then since δ(c 3 , c 4 ) < +∞, it follows immediately from the above discussion that there exists an element such that c 5 c 3 , and c 5 c 4 . Moreover, since c 1 = c 2 , and c 5 ∈ VE( X) c 3 = c 4 , by permuting {c 3 , c 4 } or {c 1 , c 2 } if necessary, we may assume without loss of generality that c 5 = c 3 , and c 3 = c 1 . On the other hand, the resulting 47 nontriviality of the specialization relations c 5 c 3 c 1 then contradicts the 1-dimensionality of the special ﬁbers of the compactiﬁed semistable models “Z” This completes the proof of assertion that appear in the deﬁnition of VE( X). (ix). def Next, we verify assertion (x). Write c 1 = c. Suppose that c 2 c 1 , c 2 c 1 \ {c 1 }. Then it follows that δ(c 2 , c ) < for distinct elements c 2 , c 2 ∈ VE( X) 2 +∞. Thus, we conclude from Proposition 2.3, (ix), that there exists an element such that c 3 c 2 , and c 3 c . In particular, by permuting c 3 ∈ VE( X) 2 {c 2 , c 2 } if necessary, we may assume without loss of generality that c 3 = c 2 , and c 2 = c 1 . On the other hand, the resulting nontriviality of the specialization relations c 3 c 2 c 1 then contradicts the uniqueness portion of Proposition 2.3, (ix). This completes the proof of assertion (x). Next, we verify assertion (xi). Let Z → X be a [connected] ﬁnite étale Ga- → Z → X such that Z has split lois covering equipped with a factorization X ∗ stable reduction over K; Z a compactiﬁed semistable model with split reduc- tion of Z over O K that is stabilized by the natural action of Gal(Z/X). [Note that it follows immediately from Proposition 2.3, (iii), that such compactiﬁed semistable models form a directed inverse system that is coﬁnal in the directed inverse system that appears in the deﬁnition of VE( X).] Write Z for the com- pactiﬁed stable model with split reduction of Z over O K ; Γ for the dual graph of Z s ; Γ ∗ for the dual graph of Z s ∗ . Observe that the natural action of s(H) on Γ ∗ factors through a ﬁnite quotient of s(H). Thus, it follows immediately from [CbTpIV], Corollary 1.15, (iii), that the natural action of s(H) on Γ has a ﬁxed point c ∈ Γ. On the other hand, it follows immediately from the well-known theory of stable and semistable models [i.e., which may be reduced, by adding ﬁnitely many suitably positioned cusps, to the theory of pointed stable curves and contraction morphisms that arise from eliminating cusps, as exposed in [Knud]] that the inverse image of [the node or interior of an irreducible compo- nent in Z s ∗ corresponding to] c via the dominant morphism Z ∗ → Z determines a tree inside Γ ∗ . Moreover, we recall that any action of a ﬁnite group on a tree has a ﬁxed point [cf., e.g., [SemiAn], Lemma 1.8, (ii)]. Thus, we conclude that the natural action of s(H) on Γ ∗ has a ﬁxed point. Since any inverse limit of nonempty ﬁnite sets is nonempty, we thus conclude that the natural action of has a ﬁxed point ∈ VE( X), hence from Proposition 2.3, (x), s(H) on VE( X) prim . that the natural action of s(H) on VE( X) prim has a ﬁxed point ∈ VE( X) This completes the proof of assertion (xi). Next, we verify assertion (xii). Fix a prime number l ∈ Σ\{p}. Then observe that it follows from the stable reduction theorem [cf. [DM], [Knud]] that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, we may assume without loss of generality that Σ = {p, l}, and that X has stable reduction over K. Write X for the [unique, up to unique isomorphism] compactiﬁed stable model of X over O K . Next, observe that it follows immediately from Hurwitz’s formula, together with the well-known structure of geometric fundamental groups of hyperbolic curves over ﬁelds of characteristic zero [cf., e.g., [CmbGC], Remark 1.1.3], that, 48 after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a connected geometrically pro-p ﬁnite étale covering Y → X of hyperbolic curves with split stable reduction over K that satisﬁes the condition that Y is of genus g Y ≥ 2. Write Y † for the smooth compactiﬁcation of Y over K; Y † for the [unique, up to unique isomorphism] compactiﬁed stable model of Y † over O K . Then observe that, if Y † is smooth over O K , then it follows immediately from the non-injectivity of the natural surjective homomorphism ab Π ab Y † ⊗ Z/pZ Π Y † ⊗ Z/pZ s [where we recall that, since Y † is of genus g Y † ≥ 2, the domain of this homo- morphism is of cardinality p 2g Y † , while the codomain of this homomorphism is of cardinality ≤ p g Y † ], together with Hurwitz’s formula [cf. also Zariski-Nagata purity; [ExtFam], Theorem A], that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a [connected] ﬁnite étale cyclic covering Y ‡ → Y † of hyperbolic curves over K that is of degree p and, moreover, satisﬁes the property that Y ‡ has bad reduction. In particular, by replacing Y ‡ × Y † Y by Y , we may assume without loss of generality that Y has bad reduction. More- over, by replacing the connected geometrically pro-p ﬁnite étale covering Y → X by its Galois closure, we may assume without loss of generality that Y → X is a [connected] geometrically pro-p ﬁnite étale Galois covering [cf. Remark 2.1.3; Hurwitz’s formula]. Next, observe that it follows immediately from the theory of admissible cov- erings [cf., e.g., [Hur], §3], together with Hurwitz’s formula [and the well-known structure of geometric pro-l fundamental groups of hyperbolic curves over ﬁelds of characteristic p = l — cf., e.g., [CmbGC], Remark 1.1.3], that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a [connected] geometrically pro-l ﬁnite étale Galois covering Z → Y of hyperbolic curves with split stable reduction over K that satisﬁes the condition that every irreducible component of the special ﬁber of the [unique, up to unique isomorphism] sta- ble model of Z is a smooth curve of genus ≥ 2. Here, note that, by replacing Z → Y by the composite of the Gal(Y /X)-conjugates of the admissible covering Z → Y , we may assume without loss of generality that the composite covering def Z → Y → X is Galois. Thus, by taking X † = Z, we obtain a [connected] geometrically pro-Σ ﬁnite étale Galois covering X † → X of hyperbolic curves satisfying the conditions in the statement of assertion (xii), as desired. This completes the proof of assertion (xii), hence of Proposition 2.3. Remark 2.3.1. We maintain the notation of Proposition 2.3. Then we observe that the statement of Proposition 2.3, (ii), becomes false if one omits the con- dition that the p-valuation v is residue-transcendental. Indeed, it suﬃces to construct an example of a discrete p-valuation on K(X) whose residue ﬁeld is algebraic over the residue ﬁeld of O K . Suppose that no ﬁnite extension ﬁeld of the residue ﬁeld of O K is separably closed [a condition that is satisﬁed if, for instance, the residue ﬁeld of O K is ﬁnite]. Then one veriﬁes immediately that 49 ur ) \ X(K) = ∅, and that for any x ∈ X( K ur ) \ X(K), the point-theoretic X( K valuation associated to x on K(X) satisﬁes the desired properties. Remark 2.3.2. An alternative proof of Proposition 2.3, (iv), may be found in [Ray2], Proposition 5. The proof of Proposition 2.3, (iv), given in the present paper is of interest in that it involves techniques that are closer to the overall approach of the present paper. ∼ an → tor of Proposition 2.3, (viii), Remark 2.3.3. The homeomorphism X VE( X) is essentially the same as the homeomorphism of [Lpg1], Proposition 1.1, but we give [essentially] self-contained statements and proofs here in the language of the present discussion for the sake of completeness. Remark 2.3.4. We maintain the notation of Proposition 2.3. Let X be a com- pactiﬁed semistable model of X over O K ; x ∈ X a smooth closed point. Write η for the generic point of the unique irreducible component of X s that contains x. Then one may construct a p-valuation v on K(X) associated to x by taking the ring of integers O v to consist of the elements ∈ K(X) that are integral with respect to the discrete valuation on K(X) associated to η and, moreover, map to an element in the residue ﬁeld k(η) of X at η that is integral with respect to the discrete valuation on k(η) determined by x. Note that η determines a prime ideal of O v that contains p. In particular, v is nonprimitive. Proposition 2.4 (First properties of resolution of nonsingularities). Let Σ ⊆ Primes be a nonempty subset; K a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; X a hyperbolic curve over K. Then: (i) Let U ⊆ X be an open subscheme [so U is a hyperbolic curve over K]. Suppose that X satisﬁes Σ-RNS. Then it holds that U satisﬁes Σ-RNS. (ii) Let f : Y → X be a connected geometrically pro-Σ ﬁnite étale covering over K [so Y is a hyperbolic curve over a ﬁnite extension ﬁeld of K]. Then it holds that X satisﬁes Σ-RNS if and only if Y satisﬁes Σ-RNS. (iii) Suppose that X satisﬁes the following condition: Let X be a compactiﬁed model of X over O K ; x ∈ X s a closed point. Then, after possibly replacing K by a suitable ﬁnite ex- tension ﬁeld of K, there exist • a connected geometrically pro-Σ ﬁnite étale Galois covering Y → X of hyperbolic curves over K, • a compactiﬁed semistable model Y of Y over O K , • a morphism Y → X of compactiﬁed models over O K that restricts to the ﬁnite étale Galois covering Y → X, 50 • an irreducible component D of Y s whose normalization is of genus ≥ 1, and whose image in X s is x ∈ X s . Then X satisﬁes Σ-RNS. (iv) Suppose that X satisﬁes Σ-RNS. Let X be a compactiﬁed model of X over O K . Then, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a connected geometrically pro-Σ ﬁnite étale Galois covering Y → X over K, together with a compactiﬁed stable model Y of Y over O K , such that the covering Y → X extends to a morphism Y → X . (v) Suppose that we are in the situation of Proposition 2.3, (v), and that X satisﬁes Σ-RNS. Write st , VE( X) st,tor , VE( X) st,prim , VE( X) st,pt-th VE( X) VE( X) tor , VE( X) prim , VE( X) pt-th for the modiﬁed versions of VE( X), obtained by requiring that the compactiﬁed semistable models “Z” that appear in the inverse limits used to deﬁne these sets be compactiﬁed stable models. [Here, we observe that, in light of (iv), the various toral compactiﬁed semistable models “Z † ” relative to “Z” that appear in the construction of “VE(Z) tor ” in Deﬁnition 2.2, (vi), may be understood as being obtained as the result of contracting suitable irreducible components in the special ﬁbers [cf. Remark 2.1.6] of suitable quotients of compactiﬁed stable models as in Proposition 2.3, (iv).] Then the natural maps −→ VE( X) st VE( X) tor −→ VE( X) st,tor VE( X) prim −→ VE( X) st,prim VE( X) pt-th −→ VE( X) st,pt-th VE( X) are bijective. (vi) Suppose that we are in the situation of Proposition 2.3, (v). Then X it holds that satisﬁes Σ-RNS if and only if for each c ∈ VE( X), R c = lim O Z st ,z c , −→ st Z denotes the valuation ring of the p-valuation associ- where R c ⊆ K( X) ated to c [cf. Proposition 2.3, (v)]; the direct limit ranges over the set of compactiﬁed stable models with split reduction Z st of the domain curves of connected ﬁnite étale coverings Z → X equipped with a factorization → Z → X; z c denotes the center on Z st determined by R c ; the tran- X sition maps in the direct limit are induced by the corresponding scheme- theoretic morphisms of compactiﬁed stable models [which, in light of (iv), form a directed inverse system]. 51 (vii) Suppose that we are in the situation of Proposition 2.3, (v), and that X satisﬁes Σ-RNS. Let l ∈ Σ \ {p}; H ⊆ G K a closed subgroup such that the intersection H ∩ I K of H with the inertia subgroup I K of G K (Σ) admits a surjection to [the proﬁnite group] Z l ; s : H → Π X a section (Σ) of the restriction to H of the natural surjection Π X G K . Then there prim that is ﬁxed by the restriction, exists at most one element c ∈ VE( X) (Σ) prim ⊆ VE( X); if, via s, to H of the natural action of Π X on VE( X) moreover, the restriction to H of the l-adic cyclotomic character of K prim . In has open image, then there exists a unique such element c ∈ VE( X) an particular, if X is proper, then there exists at most one element c an ∈ X [cf. Proposition 2.3, (vii)] that is ﬁxed by the restriction, via s, to H of (Σ) an ; if, the natural action of Π X on the topological pro-Berkovich space X moreover, the restriction to H of the l-adic cyclotomic character of an . K has open image, then there exists a unique such element c an ∈ X Proof. Assertions (i), (ii) follow immediately from the various deﬁnitions in- volved. Next, we verify assertion (iii). Let v be a discrete residue-transcendental p-valuation on K(X). Then it follows immediately from Proposition 2.3, (ii), (iii), that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a compactiﬁed semistable model X of X over O K such that v arises from an irreducible component of X s . Let {x 1 , . . . , x N } ⊆ X s be a ﬁ- nite set of distinct closed points in the smooth locus of X s such that every irreducible component of X s whose normalization is of genus 0 contains three points ∈ {x 1 , . . . , x N }. Then it follows immediately from our assumption on X that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, for each positive integer i ≤ N , there exist • a connected geometrically pro-Σ ﬁnite étale Galois covering Y i → X over K, • a morphism f i : Y i → X of compactiﬁed semistable models over O K that restricts to the ﬁnite étale Galois covering Y i → X, • an irreducible component D i of (Y i ) s whose normalization is of genus ≥ 1, and whose image in X s is x i . Write f η : Y → X for the connected geometrically pro-Σ ﬁnite étale Galois covering over K obtained by forming the composite of the ﬁnite étale Galois coverings {Y i → X} 1≤i≤N over K. Then it follows immediately from Propo- sition 2.3, (iii), that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a compactiﬁed semistable model Y † of Y over O K that dominates the respective normalizations of the semistable models {Y i } 1≤i≤N in the function ﬁeld of Y . In particular, for each positive integer i ≤ N , there exists an irreducible component D i † of (Y † ) s whose normalization is of genus ≥ 1, and whose image in X s is x i . Next, let us observe that it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that, after possibly replacing K by a suit- able ﬁnite extension ﬁeld of K, we may regard Y † as the compactiﬁed stable 52 model associated to the hyperbolic curve Y † obtained by removing from Y a collection of K-rational points of Y . In a similar vein, it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that, after pos- sibly replacing K by a suitable ﬁnite extension ﬁeld of K and replacing Y † by a suitable Gal(Y /X)-stable dense open subscheme of Y † , we may assume without loss of generality that Y † is stabilized by the action of Gal(Y /X). Write f † : Y † → X for the natural dominant morphism that restricts to the ﬁnite étale Galois cov- ering f η : Y → X; κ † : Y † → Y st for the natural dominant morphism to a compactiﬁed stable model Y st of Y over O K [cf. [ExtFam], Theorem A]. Thus, for each positive integer i ≤ N , f † (D i † ) = x i ∈ X s . In particular, since the covering f η : Y → X is Galois, and Y † is stabilized by the action of Gal(Y /X), it follows immediately from Zariski’s Main Theorem that, for each positive integer i ≤ N , the inverse image (f † ) −1 (x i ) ⊆ Y s † is a closed subscheme that contains D i † and is pure of dimension 1. Here, we recall that D i † is an irreducible component of Y s † whose normalization is of genus ≥ 1, hence necessarily maps birationally, via κ † , to an irreducible component of Y st . In particular, we conclude that each connected component of (f † ) −1 (x i ) ⊆ Y s † contains an irreducible component of Y s † that maps birationally, via κ † , to an irreducible component of Y st . Next, let D † ⊆ Y s † be an irreducible component of Y s † that maps to a closed point κ † (D † ) of Y s st via κ † : Y † → Y st , but dominates an irreducible component def E = f † (D † ) of X s . Note that these assumptions imply that the normalization of D † is of genus 0, and hence that E is an irreducible component of X s whose normalization is of genus 0. Thus, we conclude [cf. the condition imposed on the subset {x 1 , . . . , x N } ⊆ X s ] that E contains three points ∈ {x 1 , . . . , x N }, i.e., [since (f † ) −1 (x i ) is pure of dimension 1] that D † contains at least 3 nodes [that map to three distinct “x i ”]. On the other hand, this [together with the birationality of κ † ] implies that the closed point κ † (D † ) of Y s st intersects three distinct irreducible components of Y s st [i.e., the images of suitable irreducible components of (f † ) −1 (x i ) ⊆ Y s † , for three distinct “i”], that is to say, in con- tradiction to the deﬁnition of the notion of a compactiﬁed stable model [cf. Deﬁnition 2.1, (iv)]. Thus, we conclude that there do not exist any such “D † ” [i.e., that map to a closed point of Y s st , but dominate an irreducible component of X s ], and hence, by Zariski’s Main Theorem, that the morphism f † : Y † → X factors as the composite of κ † with a morphism f st : Y st → X . In particular, it follows from the existence of the morphism f st : Y st → X that Y s st contains an irreducible component whose corresponding valuation induces the given val- uation v on K(X), i.e., that X satisﬁes Σ-RNS. This completes the proof of assertion (iii). Next, we verify assertion (iv). In light of Proposition 2.3, (iii), after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, we may assume without loss 53 of generality that X is a compactiﬁed semistable model with split reduction of X over O K . Write {v 1 , . . . , v N } for the set of discrete valuations on K(X) that arise from the irreducible components of X s . Then since X satisﬁes Σ-RNS, for each positive integer i ≤ N , there exists a connected geometrically pro-Σ ﬁnite étale Galois covering Y i → X such that v i coincides with the restriction of a discrete valuation on the function ﬁeld of Y i that arises from an irreducible component of the special ﬁber of a compactiﬁed stable model of Y i . Write Y → X for the composite covering of the connected geometrically pro-Σ ﬁnite étale Galois coverings {Y i → X} 1≤i≤N . Then it follows immediately from Zariski’s Main Theorem [cf. also [ExtFam], Theorem A] that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, there exists a compactiﬁed stable model Y of Y over O K that dominates X . This completes the proof of assertion (iv). Assertion (v) and the necessity portion of assertion (vi) follow immediately from Proposition 2.4, (iv), together with Proposition 2.3, (iv), (vi). Next, we be an element consider the suﬃciency portion of assertion (vi). Let c ∈ VE( X) that extends that corresponds [cf. Proposition 2.3, (v)] to a valuation v on K( X) a discrete residue-transcendental p-valuation on X. [Note that in this situation, v itself is necessarily residue-transcendental.] Then it suﬃces to show that there exists a “z c ” as in the statement of Proposition 2.4, (vi), that is a generic point of “Z s st ”. To this end, we observe that the nonexistence of such a “z c ” would imply that all of the “z c ” are closed points of “Z s st ”, hence have residue ﬁelds that are algebraic over the residue ﬁeld of O K . On the other hand, this would imply that the residue ﬁeld of R c = O v is algebraic over the residue ﬁeld of O K , in contradiction to the residue-transcendentality of v. This completes the proof of assertion (vi). Finally, we verify assertion (vii). The portion of assertion (vii) concerning prim as in the statement of Proposition the existence of an element c ∈ VE( X) 2.4, (vii), follows from Proposition 2.3, (xi). To verify the portion of assertion prim , it suﬃces (vii) concerning the uniqueness of such an element c ∈ VE( X) prim that satisfy the to show the equality of any two elements c 1 , c 2 ∈ VE( X) condition imposed on element “c” in the statement of Proposition 2.4, (vii). st,prim . Then it follows from Write c 1 , c 2 for the images of c 1 , c 2 in VE( X) Proposition 2.4, (iv); [CbTpIV], Corollary 1.15, (iv) [applied to c 1 , c 2 ], that δ(c 1 , c 2 ) < +∞, hence from Proposition 2.3, (ix), that c 1 = c 2 , as desired. This completes the proof of assertion (vii), hence of Proposition 2.4. Corollary 2.5 (Constructions associated to geometric tempered fun- damental groups). Let Σ ⊆ Primes be a subset of cardinality ≥ 2 such that p ∈ Σ; K † , K ‡ mixed characteristic complete discrete valuation ﬁelds of residue † ‡ characteristic p; X † , X ‡ hyperbolic curves over K , K , respectively. Write Ω † , † ‡ Ω ‡ for the p-adic completions of K , K , respectively. For any hyperbolic curve † ‡ Z over K , K , Ω † , or Ω ‡ , write Π tp Z for the Σ-tempered fundamental group of Z, relative to a suitable choice of basepoint [cf. the subsection in Notations 54 † → X † , X ‡ → X ‡ and Conventions entitled “Fundamental groups”]. Write X tp for the universal geometrically pro-Σ coverings corresponding to Π X † , Π tp , re- X ‡ spectively. Suppose that X † and X ‡ satisfy Σ-RNS. Then the following hold [cf. Remark 2.5.1 below]: ∼ (i) Let σ : Π tp → Π tp be an isomorphism of topological groups. Then σ X † X ‡ induces homeomorphisms ∼ ‡ ), † ) → VE( X VE( X ∼ † ) tor → ‡ ) tor , VE( X VE( X ∼ † ) prim → VE( X ‡ ) prim , VE( X that are compatible with the respective natural actions of Π tp , Π tp . If, X † X ‡ † ‡ moreover, X and X are proper, then σ induces a homeomorphism ∼ ‡ ) an † ) an → ( X ( X that is compatible with the respective natural actions of Π tp , Π tp . X † X ‡ † def ‡ def (ii) Suppose that K = K † = K ‡ , K = K = K , hence that Ω = Ω † = Ω ‡ . Let x † ∈ X † (Ω), x ‡ ∈ X ‡ (Ω). Write X x † † (respectively, X x ‡ ‡ ) for the ∼ : Π tp † → hyperbolic curve X Ω † \{x † } (respectively, X Ω ‡ \{x ‡ }) over Ω. Let σ X † x tp Π ‡ be an isomorphism of topological groups that ﬁts into a commutative X ‡ x diagram ∼ Π tp † −−−−→ Π tp ‡ X † X ‡ σ ⏐ x ⏐ x ⏐ ⏐ ∼ Π tp −−−−→ Π tp , X † X ‡ σ where the vertical arrows are the natural surjections [determined up to composition with an inner automorphism] induced by the natural open immersions X x † † → X Ω † , X x ‡ ‡ → X Ω ‡ of hyperbolic curves; the lower hori- zontal arrow σ is the isomorphism of topological groups [determined up to composition with an inner automorphism] induced by a(n) [uniquely de- ∼ termined — cf., e.g., [DM], Lemma 1.14] isomorphism σ X : X † → X ‡ of schemes over K. Then x ‡ = σ X (x † ). Proof. First, we verify assertion (i). We begin by recalling that [SemiAn], Corol- lary 3.11, may be generalized/applied to hyperbolic curves over an arbitrary mixed characteristic complete discrete valuation ﬁeld [cf. [AbsTopII], Remark 2.11.1, (i)]. Thus, by applying this generalized version of [SemiAn], Corollary ∼ ‡ ) st . † ) st → VE( X 3.11, we conclude that σ induces a homeomorphism VE( X † ‡ On the other hand, it follows from our assumption that X and X satisfy ∼ → st ” and Σ-RNS that we may apply the homeomorphisms “VE( X) VE( X) 55 ∼ tor → st,tor ” of Proposition 2.4, (v). In particular, we conclude “VE( X) VE( X) that σ induces a homeomorphism ∼ ‡ ) tor † ) tor → VE( X VE( X that is manifestly compatible with the respective natural actions of Π tp , Π tp , X † X ‡ as well as a homeomorphism ∼ ‡ ) † ) → VE( X VE( X , Π tp that is manifestly compatible with the respective natural actions of Π tp X † X ‡ and preserves specialization/generization relations, hence induces a homeomor- phism ∼ † ) prim → ‡ ) prim VE( X VE( X that is compatible with the respective natural actions of Π tp , Π tp . Finally, if, X † X ‡ † ‡ moreover, X and X are proper, then we may apply the homeomorphism “θ X ” of Proposition 2.3, (viii), to conclude that σ induces a homeomorphism ∼ † ) an → ‡ ) an ( X ( X that is compatible with the respective natural actions of Π tp , Π tp . This com- X † X ‡ pletes the proof of assertion (i). Next, we verify assertion (ii). We begin by observing that it follows from the generalized version of [SemiAn], Corollary 3.11, discussed above, together with Corollary 2.5, (i), that σ induces a bijection ∼ † ) → ‡ ) VE( X VE( X that maps the point-theoretic orbit-center-chain associated to x † to the point- theoretic orbit-center-chain associated to x ‡ . Since σ arises from σ X , we thus ∼ pt-th /Gal( X/X conclude from the bijection “X(Ω) → VE( X) K )” of Proposition † ‡ 2.3, (v), that σ X (x ) = x . This completes the proof of assertion (ii), hence of Corollary 2.5. ∼ † ) an → ‡ ) an ” of Corollary 2.5, (i), Remark 2.5.1. The homeomorphism “( X ( X is essentially similar to the homeomorphisms of [Lpg1], Theorem 3.10, but is formulated and proven according to the approach of the present paper. On the other hand, Corollary 2.5, (ii), may be regarded, when taken together with Theorem 2.17 below, as a generalization of [Tsjm], Theorem 2.2; its proof may be regarded as a more sophisticated version of the argument applied in the proof of [Tsjm], Theorem 2.2. Proposition 2.6 (Existence of new ordinary parts of certain cover- ings after Raynaud-Tamagawa). Let l be a prime number = p; K a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; X a 56 proper hyperbolic curve over K. Suppose that X has split stable reduction over K. Write X st for the [unique, up to unique isomorphism] stable model of X over O K . Suppose, moreover, that every irreducible component of the special ﬁber of X st is a smooth curve of genus ≥ 2. Write e X (respectively, v X ) for the cardinality of the set of nodes (respectively, the set of irreducible components) of the stable curve X s st . Then: (i) For each suﬃciently large positive integer m, if we replace K by a ﬁnite unramiﬁed extension ﬁeld of K, then there exists a ﬁnite étale cyclic cov- ering Y st −→ X st over O K of degree l m satisfying the following conditions: (a) Write (Y st →) Z st → X st for the ﬁnite étale cyclic subcovering over O K of degree l m−1 ; Y , Z for the generic ﬁbers of Y st , Z st , respec- tively. Then Y and Z have split stable reduction over K. Moreover, Y st , Z st are the stable models of Y , Z, respectively. (b) The ﬁnite étale covering Y s st → X s st determined by the ﬁnite étale cyclic covering Y st → X st induces a bijection between the respective sets of irreducible components. (c) Write A for the abelian variety over K obtained by forming the coker- nel of the natural morphism J(Z) → J(Y ) induced by the ﬁnite étale cyclic covering Y → Z [of degree l]. Then there exists an abelian va- riety B over K with good ordinary reduction such that T p A ﬁts into exact sequences of G K -modules [cf. the theory of [FC], especially, [FC], Chapter III, Corollary 7.3] 0 −→ T gd −→ T p A −→ T cb −→ 0 0 −→ T tor −→ T gd −→ T p B −→ 0 0 −→ Hom(T p B s , Z p (1)) −→ T p B −→ T p B s −→ 0, where “(1)” denotes the Tate twist; the natural action of G K on the “combinatorial quotient” T cb [i.e., the inverse limit of the quotients “Y /nY ” of [FC], Chapter III, Corollary 7.3, as n ranges over the positive integral powers of p] of T p A is trivial; T tor is isomorphic as a G K -module to the direct sum of a collection of copies of Z p (1); B denotes the abelian scheme over O K whose generic ﬁber is equal to B. (ii) Fix a ﬁnite étale cyclic covering Y st → X st as in (i). Write T cb,Y , T cb,Z for the “combinatorial quotients” [i.e., the inverse limit of the quotients “Y /nY ” of [FC], Chapter III, Corollary 7.3, as n ranges over the positive integral powers of p] of T p J(Y ), T p J(Z), respectively; h Y , h Z for the 57 respective loop-ranks of the dual graphs associated to the stable curves Y s st , Z s st . Then it holds that h Y = 1 + l m e X − v X , h Z = 1 + l m−1 e X − v X . Moreover, rank Z p T cb,Y = h Y ; rank Z p T cb,Z = h Z ; rank Z p T tor = rank Z p T cb = h Y − h Z = (l m − l m−1 )e X . Proof. First, we verify assertion (i). Write {C i } 1≤i≤v X for the set of irreducible components of X s st . Let m be a positive integer such that, for each positive integer i ≤ v X , it holds that l m > l 2g Ci − l 2g Ci −1 (p − 1)g C i , l 2g Ci − 1 where g (−) denotes the genus of (−). Then, in light of [Tama1], Lemma 1.9 [i.e., a generalization of [Ray1], Théorème 4.3.1], by replacing K by a ﬁnite unramiﬁed extension ﬁeld of K, one may construct ﬁnite étale cyclic coverings {D i → C i } 1≤i≤v X of degree l m [of proper hyperbolic curves over the residue ﬁeld of K] satisfying the following conditions: • For each positive integer i ≤ v X , write (D i →) E i → C i for the ﬁnite étale cyclic subcovering of degree l m−1 [of proper hyperbolic curves over the residue ﬁeld of K]. Then the abelian variety obtained by forming the cokernel of the natural morphism J(E i ) → J(D i ) induced by the ﬁnite étale cyclic covering D i → E i of degree l is ordinary. • The cardinality of the set of closed points of D i that lie over the closed points of C i determined by the nodes of X s st is equal to l m . Next, one veriﬁes immediately that there exists a ﬁnite étale cyclic covering D → X s st of degree l m obtained by gluing together the ﬁnite étale cyclic cov- erings {D i → C i } 1≤i≤v X . Write Y st → X st for the ﬁnite étale cyclic covering obtained by deforming the ﬁnite étale cyclic covering D → X s st . Then it follows immediately from the various deﬁnitions involved that conditions (a), (b) hold. Moreover, in light of the theory of Raynaud extensions [cf. [FC], Chapter II, §1; [FC], Chapter III, Corollary 7.3], together with Remark 2.6.1, (i), (ii), below [cf. also [BLR], §9.2, Example 8], one concludes that condition (c) holds. This completes the proof of assertion (i). Next, we verify assertion (ii). Write e Y , e Z for the respective cardinalities of the sets of nodes of the stable curves Y s st , Z s st ; v Y , v Z for the respective cardinalities of the sets of irreducible components of the stable curves Y s st , Z s st . Then it follows immediately from conditions (a), (b), that e Y = l m e X , e Z = l m−1 e X , 58 v Y = v Z = v X . Thus, we conclude from the well-known computation of the ﬁrst homology group of a ﬁnite graph that h Y = 1 + e Y − v Y = 1 + l m e X − v X , h Z = 1 + e Z − v Z = 1 + l m−1 e X − v X . Therefore, to complete the proof of assertion (ii), it suﬃces to prove that rank Z p T cb,Y = h Y , rank Z p T cb,Z = h Z , and rank Z p T tor = rank Z p T cb = h Y −h Z . Recall that the loop-ranks h Y , h Z coincide with the toric ranks of the Jacobians of the stable curves Y s st , Z s st , respectively [cf., e.g, [BLR], §9.2, Example 8]. On the other hand, in light of the theory of duality for torsion subgroups of abelian varieties, it holds that these toric ranks coincide with the ranks of the respective corresponding combinatorial quotients [cf. [FC], Chapter III, Corollary 7.4]. In particular, it follows immediately [cf. Remark 2.6.1, (ii), below; [FC], Chap- ter III, Corollary 7.4] that rank Z p T cb,Y = h Y , rank Z p T cb,Z = h Z , hence that rank Z p T tor = rank Z p T cb = rank Z p T cb,Y − rank Z p T cb,Z = h Y − h Z . This completes the proof of assertion (ii), hence of Proposition 2.6. Remark 2.6.1. We maintain the notation of Proposition 2.6. (i) Write A for the identity component of the Néron model of A over O K [cf. [BLR], §1.3, Corollary 2]. Then the universal property of the Néron model implies the existence of a surjective homomorphism f : Pic 0 Y st /O K A that extends the natural quotient homomorphism J(Y ) A [cf. [BLR], §1.2, Deﬁnition 1; [BLR], §9.4, Theorem 1]. Thus, since Pic 0 Y st /O K is a semi-abelian scheme over O K [cf. [BLR], §9.4, Theorem 1], it follows immediately from the existence of the surjective homomorphism f that A is also a semi-abelian scheme over O K . (ii) Recall that the composite homomorphism J(Z) → J(Y ) → J(Z) of the norm map J(Y ) → J(Z) with the natural homomorphism J(Z) → J(Y ) coincides with the morphism given by multiplication by l. In particular, the abelian variety J(Y ) is isogenous over K to the product abelian variety J(Z) × K A. Thus, we conclude from [BLR], §7.3, Proposition 6, that the semi-abelian schemes Pic 0 Y st /O K and Pic 0 Z st /O K × O K A over O K [cf. [BLR], §9.4, Theorem 1] are isogenous over O K . Deﬁnition 2.7. In the notation of Remark 2.6.1, let Z be a semistable model of Z over O K that has split reduction. Note that Z st satisﬁes this property, and that, by pulling-back the ﬁnite étale cyclic covering Y st → Z st via the unique morphism Z → Z st that extends the identity morphism Z → Z [cf. [ExtFam], Theorem A], we obtain a ﬁnite étale cyclic covering Y −→ Z over O K of degree l that extends the ﬁnite étale cyclic covering Y → Z over K. Suppose that 59 • X s st is a singular curve, and that • m is suﬃciently large that h Y ≥ h Z ≥ 1 [cf. Proposition 2.6, (ii)]. (i) Let y η ∈ Y (K). Write z η ∈ Z(K) for the image of y η via the natural map Y (K) → Z(K). Then y η and z η determine embeddings Y → J(Y ), Z → J(Z) that allow one to regard J(Y ), J(Z) as the respective Albanese varieties of Y , Z [cf. [AbsTopI], Appendix, Deﬁnition A.1, (ii); [Milne], Proposition 6.1]. In particular, we obtain a commutative diagram Δ Y −−−−→ T p J(Y ) −−−−→ T cb,Y ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Δ Z −−−−→ T p J(Z) −−−−→ T cb,Z , where the left-hand vertical arrow denotes the open injection induced by the ﬁnite étale cyclic covering Y → Z; the left-hand horizontal arrows denote the natural surjections determined by the Albanese embeddings Y → J(Y ), Z → J(Z) [cf. [AbsTopI], Appendix, Proposition A.6, (iv)]; the right-hand horizontal arrows denote the natural surjections; the mid- dle and right-hand vertical arrows are surjections [cf. the fact that the ﬁnite étale cyclic covering Y → Z is of degree l = p]; the ﬁrst square of the diagram commutes in light of the functoriality of the étale funda- mental group; the second square of the diagram commutes in light of the functoriality of Raynaud extensions. (ii) Fix a quotient T cb,Z Z p [cf. our assumption that h Z ≥ 1; Proposition 2.6, (ii)]. For each nonneg- ative integer n, write Z n −→ Z for the ﬁnite étale cyclic [“combinatorial”] covering of degree p n over O K induced by the natural quotient (Δ Z T p J(Z) ) T cb,Z Z p Z/p n Z; Y n −→ Y for the ﬁnite étale cyclic [“combinatorial”] covering of degree p n over O K induced by the natural quotient (Δ Y T p J(Y ) ) T cb,Y T cb,Z Z p Z/p n Z. Thus, the commutative diagram in (i) induces a cartesian commutative diagram Y n −−−−→ Y ⏐ ⏐ ⏐ ⏐ Z n −−−−→ Z, 60 where the vertical arrows are ﬁnite étale cyclic coverings of degree l over O K ; the horizontal arrows are ﬁnite étale cyclic [“combinatorial”] cover- ings of degree p n over O K . Moreover: (a) Write Y n , Z n for the generic ﬁbers of Y n , Z n , respectively. Then the ﬁnite étale covering (Y n ) s → (Z n ) s determined by the ﬁnite étale cyclic covering Y n → Z n induces a bijection between the sets of irreducible components that arise from the respective stable models of Y n and Z n [cf. Proposition 2.6, (i), (b)]. (b) Write A n for the abelian variety over K obtained by forming the cokernel of the natural morphism J(Z n ) → J(Y n ) induced by the ﬁnite étale cyclic covering Y n → Z n of degree l [of proper hyperbolic curves over K]. Then there exists an abelian variety B n over K with good ordinary reduction such that T p A n ﬁts into exact sequences of G K -modules [cf. the theory of [FC], especially, [FC], Chapter III, Corollary 7.3; the proof of Proposition 2.6, (i), (c)] 0 −→ T gd,n −→ T p A n −→ T cb,n −→ 0 0 −→ T tor,n −→ T gd,n −→ T p B n −→ 0 0 −→ Hom(T p (B n ) s , Z p (1)) −→ T p B n −→ T p (B n ) s −→ 0, where “(1)” denotes the Tate twist; the natural action of G K on the “combinatorial quotient” T cb,n of T p A n is trivial; T tor,n is isomorphic as a G K -module to the direct sum of a collection of copies of Z p (1); B n denotes the abelian scheme over O K whose generic ﬁber is equal to B n . (iii) Write A n for the identity component of the Néron model of A n over O K [cf. [BLR], §1.3, Corollary 2]. Then the universal property of the Néron model implies the existence of a surjective homomorphism def f n : J n = Pic 0 Y n /O K A n that extends the natural quotient homomorphism J(Y n ) A n [cf. [BLR], §1.2, Deﬁnition 1; [BLR], §9.4, Theorem 1]. Thus, since J n = Pic 0 Y n /O K is a semi-abelian scheme over O K [cf. [BLR], §9.4, Theorem 1], it follows immediately from the existence of the surjective homomorphism f n that A n is also a semi-abelian scheme over O K [cf. Remark 2.6.1, (i)]. (iv) For each nonnegative integer n, write h Y n , 61 h Z n for the loop-ranks of the dual graphs associated to the semistable curves (Y n ) s , (Z n ) s , respectively; g Y n for the [arithmetic] genus of the semistable model Y n over O K . Then a similar argument to the argument applied in Remark 2.6.1, (ii), implies that the semi-abelian schemes Pic 0 Y n /O K and Pic 0 Z n /O K × O K A n over O K are isogenous over O K . In particular, we obtain equalities rank Z p T tor,n = rank Z p T cb,n = h Y n − h Z n [cf. the proof of Proposition 2.6, (ii)]. Proposition 2.8 (Explicit computations of toric rank and genus). We maintain the notation of Deﬁnition 2.7. Then the following hold: (i) It holds that h Y n = 1 + p n l m e X − p n v X , h Z n = 1 + p n l m−1 e X − p n v X , hence, in particular, that rank Z p T tor,n = rank Z p T cb,n = h Y n − h Z n = p n (l m − l m−1 )e X [cf. the ﬁnal display of Deﬁnition 2.7, (iv)]. (ii) It holds that g Y n = p n (g Y 0 − 1) + 1. Proof. First, recall from the well-known theory of stable and semistable models that h Y n , h Z n , rank Z p T tor,n , rank Z p T cb,n , and g Y n are independent of the choice of the semistable model Z of Z over O K . [Indeed, by passing to a suitable ﬁnite unramiﬁed extension of K and adding ﬁnitely many suitably positioned cusps, one may, in eﬀect, reduce this “well-known theory of stable and semistable models” to the theory of pointed stable curves and contraction morphisms that arise from eliminating cusps, as exposed in [Knud].] In particular, we may assume without loss of generality that Z is the stable model of Z over O K . Assertion (i) then follows immediately from a similar argument to the argument applied in the proof of Proposition 2.6, (ii). Assertion (ii) follows immediately from Hurwitz’s formula. This completes the proof of Proposition 2.8. Deﬁnition 2.9. We maintain the notation of Deﬁnition 2.7. Then we shall write T cnn,n 62 for the connected p-divisible group over O K that arises as the connected part of the p-divisible group [cf. the discussion preceding [Tate], §2.2, Proposition 2] associated to the Raynaud extension [cf. [FC], Chapter II, §1] of A n ; T tor,n for the connected p-divisible group over O K associated to the torus that appears in the Raynaud extension of A n ; T cnn,n , T tor,n for the respective generic ﬁbers of T cnn,n , T tor,n ; def T cnn,n = T p ( T cnn,n ), def T tor,n = T p ( T tor,n ) for the respective p-adic Tate modules of T cnn,n , T tor,n [cf. the subsection in Notations and Conventions entitled “Schemes”]. Note that T cnn,n and T tor,n may be regarded as Z p -submodules of T p A n in a natural way. Moreover, we shall write def T ét,n = T p A n /T cnn,n . Note that G K acts naturally on the Z p -modules T cnn,n , T tor,n , T ét,n , and T cb,n [cf. Deﬁnition 2.7, (ii), (b)]. We shall write T cnn,n T qtr,n for the maximal torsion-free G K -stable quotient Z p -module among the torsion- free G K -stable quotient Z p -modules T cnn,n T such that some open subgroup of G K acts on T via the p-adic cyclotomic character; T qcb,n ⊆ T ét,n for the maximal G K -stable Z p -submodule among the G K -stable Z p -submodules T ⊆ T ét,n such that some open subgroup of G K acts trivially on T . Finally, we observe that [one veriﬁes immediately that] we obtain natural exact sequences of G K -modules [cf. Deﬁnition 2.7, (ii), (b)] 0 −→ T tor,n −→ T cnn,n −→ Hom(T p (B n ) s , Z p (1)) −→ 0 0 −→ T p (B n ) s −→ T ét,n −→ T cb,n −→ 0 0 −→ T cnn,n −→ T p A n −→ T ét,n −→ 0. Lemma 2.10. We maintain the notation of Deﬁnition 2.9. Then: (i) The natural action of G K on T p A n induces the trivial action of I K on T ét,n . 63 (ii) There exist natural compatible G K -equivariant isomorphisms ∼ T ét,n → Hom(T cnn,n , Z p (1)), ∼ T cb,n → Hom(T tor,n , Z p (1)), ∼ T qcb,n → Hom(T qtr,n , Z p (1)). (iii) Suppose that K is a p-adic local ﬁeld. Then the set of eigenvalues of the Z p -linear automorphism of T p (B n ) s induced by the Frobenius element of G K /I K [cf. (i); the second exact sequence of Deﬁnition 2.9] does not contain any roots of unity. Proof. First, we verify assertion (i). Recall that the quotient of a p-divisible group by its connected part is étale [cf., e.g., the discussion preceding [Tate], §2.2, Proposition 2]. Thus, we conclude [cf. the triviality of the action of G K on T cb,n observed in Deﬁnition 2.7, (ii), (b); the second exact sequence of the ﬁnal display of Deﬁnition 2.9] from [the second sentence of] [FC], Chapter III, Corollary 7.3, that the natural action of I K on T ét,n is trivial, as desired. This completes the proof of assertion (i). Assertion (ii) follows immediately from the theory of duality for torsion subgroups of abelian varieties [cf. [FC], Chapter III, Corollary 7.4], together with the ﬁrst and second exact sequences of the ﬁnal display of Deﬁnition 2.9. Assertion (iii) follows immediately from the ﬁniteness of the set of rational points of (B n ) s over any ﬁnite extension ﬁeld of the [ﬁnite!] residue ﬁeld of K. This completes the proof of Lemma 2.10. Proposition 2.11 (Toral quotient of the connected part). In the notation of Deﬁnition 2.9, write χ n : T cnn,n T qtr,n for the natural surjection of G K -modules. Then the following hold: (i) Suppose that the residue ﬁeld of K is separably closed. Then T cnn,n = T qtr,n = {0}, T qcb,n = T ét,n = {0}. (ii) Suppose that K is a p-adic local ﬁeld. Then the restriction of χ n to T tor,n ⊆ T cnn,n induces an injection T tor,n → T qtr,n with ﬁnite cokernel. Proof. First, we verify assertion (i). Note that since the residue ﬁeld of K is separably closed, G K = I K . Thus, it follows immediately from Lemma 2.10, (i), together with the various deﬁnitions involved, that T qcb,n = T ét,n . On the other hand, it follows immediately from Proposition 2.8, (i) [cf. also Proposition 2.6; Deﬁnition 2.7; the second exact sequence of the ﬁnal display of Deﬁnition 2.9], that T ét,n = {0}. Thus, we conclude from Lemma 2.10, (ii), that T cnn,n = T qtr,n = {0}. This completes the proof of assertion (i). def Next, we verify assertion (ii). Write N = rank Z p T qtr,n . Note that, in light of the maximality of T qtr,n , it suﬃces to verify that there exists a unique torsion-free G K -stable quotient Z p -module T cnn,n T 64 whose restriction to T tor,n ⊆ T cnn,n induces an injection T tor,n → T with ﬁnite cokernel, and that rank Z p T ≥ N . Let Q p be an algebraic closure of Q p equipped with the trivial action of G K . Then observe that it follows from a routine argument involving Galois descent from Q p to Q p that it suﬃces to verify that there exists a unique G K -stable quotient Q p -vector space T cnn,n ⊗ Z p Q p −→ V whose restriction to T tor,n ⊗ Z p Q p ⊆ T cnn,n ⊗ Z p Q p induces an isomorphism ∼ T tor,n ⊗ Z p Q p → V , and that dim Q p V ≥ N . Write k for the ﬁnite residue ﬁeld of the p-adic local ﬁeld K [so G K /I K may be identiﬁed with the absolute Galois group G k of k]. Then, by applying Lemma 2.10, (i), (ii), we conclude that it suﬃces to verify that there exists a unique G k -stable Q p -subspace V ∗ ⊆ T ét,n ⊗ Z p Q p whose composite with the natural surjection T ét,n ⊗ Z p Q p T cb,n ⊗ Z p Q p ∼ induces an isomorphism V ∗ → T cb,n ⊗ Z p Q p , and that dim Q p V ∗ ≥ N . On the other hand, in light of the eigenspace decomposition associated to the natural action of the Frobenius element ∈ G k , the existence and uniqueness of such a subspace, together with the inequality dim Q p V ∗ ≥ N (= rank Z p T qcb,n ) [cf. Lemma 2.10, (ii)], follows immediately from Lemma 2.10, (iii) [cf. also the triviality of the action of G K on T cb,n observed in Deﬁnition 2.7, (ii), (b)]. This completes the proof of assertion (ii), hence of Proposition 2.11. Proposition 2.12 (Construction of a certain morphism of formal schemes to the quasi-toral quotient). In the notation of Proposition 2.11, let y n ∈ Y n (O K ) be an O K -rational point that maps the closed point of Spec O K to a smooth point (y n ) s of the semistable curve (Y n ) s . Write y n,η ∈ Y n (K) for the K-valued point of Y n determined by y n ∈ Y n (O K ); C ⊆ (Y n ) s for the unique irreducible component that contains (y n ) s ; F ⊆ Y n for the closed subset obtained by forming the union of the irreducible components = C of (Y n ) s ; U y n ⊆ Y n for the open subscheme obtained by forming the complement of F ⊆ Y n ; h n,η : Y n −→ J(Y n ) for the Albanese map that maps y n,η to the origin [cf. [AbsTopI], Appendix, Deﬁnition A.1, (ii); [Milne], Proposition 6.1]. Recall that J n is a semi-abelian scheme over O K [cf. Deﬁnition 2.7, (iii)] whose generic ﬁber is J(Y n ). In particular, J n is isomorphic to the identity component of the Néron model over O K of J(Y n ) [cf. [BLR], §7.4, Proposition 3]. Thus, since U y n is a connected smooth scheme over O K whose generic ﬁber is Y n , the universal property of the Néron model implies the existence of a unique morphism h n : U y n −→ J n that extends h n,η . Next, write J n , A n for the formal completions at the origin of Y ,y , O U ,y for the completions the semi-abelian schemes J n , A n over O K ; O n n yn n 65 at y n of O Y n ,(y n ) s , O U yn ,(y n ) s . Then the natural composite map Y ,y = Spf O U ,y −→ J n −→ A n Spf O n n yn n induced by h n and the surjective homomorphism f n : J n A n [cf. Deﬁnition 2.7, (iii)] determines a morphism of formal O K -schemes Y ,y −→ T cnn,n , Spf O n n where we regard the connected p-divisible group T cnn,n as a formal group over O K [cf. [Tate], §2.2, Proposition 1]. In particular, by forming the composite with the morphism of formal O K -schemes induced by χ n [cf. [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1], we obtain a morphism of formal O K -schemes Y ,y −→ T qtr,n , Spf O n n where T qtr,n denotes the formal group over O K determined by the connected p- divisible group associated to the G K -module T qtr,n [cf. [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1]. Proof. Proposition 2.12 follows immediately from the various references quoted in the statement of Proposition 2.12. Proposition 2.13 (Coverings associated to characters). We maintain the notation of Proposition 2.12. Then the following hold: m for the formal completion at the origin of the multiplicative (i) Write G m ) for the Z p -module of group scheme G m over O K ; Hom O K ( T qtr,n , G homomorphisms over O K from T qtr,n to G m ; Hom G K (T qtr,n , Z p (1)) for the Z p -module of G K -equivariant homomorphisms of Z p -modules T qtr,n → Z p (1). Then the natural homomorphism m ) −→ Hom G (T qtr,n , Z p (1)) Hom O K ( T qtr,n , G K is bijective. m ). Consider the com- (ii) Let a be a positive integer; f ∈ Hom O K ( T qtr,n , G posite ∼ m Y ,y −→ G (Spf O K [[t]] →) Spf O n n [where t is an indeterminate, and we regard O K [[t]] as being equipped with the t-adic topology] of the morphism in the ﬁnal display of Proposition 2.12 with f . By a slight abuse of notation, we shall also write f ∈ O K [[t]] × m via the homomorphism for the image of the canonical coordinate U of G of rings induced by the above composite morphism. Then the covering of Spf O K [[t]] obtained by extracting a p a -th root of f is dominated by the covering of Spf O K [[t]] obtained by restricting the covering determined by multiplication by p a on A n . 66 Proof. Assertion (i) follows immediately from [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1. Assertion (ii) follows immediately from the various deﬁnitions involved. This completes the proof of Proposition 2.13. Lemma 2.14. Let X be a smooth proper curve of genus g X over a ﬁeld K; x ∈ X(K) a K-valued point of X; d x a nonnegative integer such that d x ≥ 2g X − 1. Then the natural composite map H 0 (X, Ω X ) → Ω X,x Ω X,x /m d x x Ω X,x is injective. Proof. First, observe that since d x ≥ 2g X − 1 > 2g X − 2 [which implies that the degree of the line bundle Ω X (−d x ·x) is negative], it follows that H 0 (X, Ω X (−d x · x)) = 0. Thus, the desired injectivity follows immediately by applying the [left exact] functor H 0 (X, −) to the short exact sequence 0 −→ Ω X (−d x ) −→ Ω X −→ Ω X,x /m d x x Ω X,x −→ 0. This completes the proof of Lemma 2.14. Lemma 2.15. Let a, b be positive integers; K a p-adic local ﬁeld of degree def d K = [K : Q p ] over Q p ; W ⊆ K[T ]/(T b+1 ) a Q p -vector subspace of dimension a. For each nonzero element h i T i ∈ K[T ]/(T b+1 ), h = 0≤i≤b def write ord(h) (≤ b) for the smallest integer i such that h i = 0. Set ord(0) = +∞. Suppose that def W \{0} ⊆ F = {h ∈ K[T ]/(T b+1 ) | ord(h) = p j −1 for some nonnegative integer j}. Then it holds that a ≤ d K (log p (b + 1) + 1) d K (b + 1) = dim Q p K[T ]/(T b+1 ) . Proof. For each nonnegative integer j, write def F j = W ∩ {h ∈ K[T ]/(T b+1 ) | ord(h) ≥ p j − 1} [so F j is a Q p -vector space, and F j+1 ⊆ F j ]. Then it follows immediately from our assumption that W \ {0} ⊆ F that for each nonnegative integer j, dim Q p (F j /F j+1 ) ≤ dim Q p (K) = d K . 67 On the other hand, since F j = {0} for any nonnegative integer j such that p j > p j − 1 ≥ b + 1, we thus conclude that a = dim Q p (W ) = +∞ dim Q p (F j /F j+1 ) ≤ d K (log p (b + 1) + 1), j=0 as desired. This completes the proof of Lemma 2.15. Theorem 2.16 (Existence of suitable coverings). In the notation of Propo- sition 2.13, suppose further that K is a p-adic local ﬁeld. Then there exists a def def real number C g Y ,d K that depends only on g Y = g Y 0 and d K = [K : Q p ] such that for any positive integer n ≥ C g Y ,d K , after possibly replacing K by a ﬁnite extension ﬁeld of K, there exist • a [connected] ﬁnite étale Galois covering W n → Y n of proper hyperbolic curves over K of degree a power of p, • a semistable model W n of W n over O K , • a morphism ψ : W n → Y n of semistable models over O K that restricts to the ﬁnite étale Galois covering W n → Y n , • an irreducible component D of (W n ) s whose normalization is of genus ≥ 1 such that ψ(D) = (y n ) s ∈ (Y n ) s . Proof. Fix a positive integer n. First, we consider the natural homomorphisms of Z p -modules ∼ Hom G K (T qtr,n , Z p (1)) ← m ) → H 0 ( T qtr,n , Ω Hom O K ( T qtr,n , G inv T qtr,n ) 0 ( T cnn,n , Ω T cnn,n ) → H inv ∼ ← H 0 (A n , Ω A n ) → H 0 (U y n , Ω Y n ) Y ,y → Ω Y n ,y n ⊗ O Y n ,yn O n n 2g Yn → Ω Y n ,y n,η /m y n,η Ω Y n ,y n,η ∼ 2g Yn → K[t]/(t ) dt , where • the ﬁrst arrow denotes the natural bijective homomorphism of Proposition 2.13, (i); 68 0 • “H inv (−)” denotes the O K -submodule of “H 0 (−)” that consists of the invariant diﬀerentials on the p-divisible group in the ﬁrst argument of “H 0 (−)”; • the second arrow denotes the injection obtained by pulling back the in- def variant diﬀerential d log(U ) = dU U on G m ; • the third arrow denotes the injection induced by χ n [cf. Proposition 2.11; [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1]; • the fourth arrow denotes the natural isomorphism; • the ﬁfth arrow denotes the homomorphism of O K -modules obtained by pulling back the diﬀerentials via the composite map U y n → J n A n that maps y n to the origin [cf. Deﬁnition 2.7, (ii), (b); Deﬁnition 2.7, (iii); Proposition 2.12]; • the sixth arrow denotes the natural injection; • the seventh arrow denotes the natural restriction morphism; • m y n,η denotes the maximal ideal of O Y n ,y n,η ; • the ﬁnal arrow denotes the natural isomorphism determined by choosing a “local coordinate” t, i.e., an element of the maximal ideal m Y n ,y n of O Y n ,y n such that t and m K generate m Y n ,y n . Write m ) −→ O K [[t]] × Ψ : Hom O K ( T qtr,n , G for the assignment discussed in Proposition 2.13, (ii) [i.e., relative to the local coordinate t chosen above]; ∼ Yn m ) → Ω Y ,y /m 2g → K[t]/(t 2g Yn ) dt Ξ : Hom O K ( T qtr,n , G y n,η Ω Y n ,y n,η n n,η for the injective [by Lemma 2.14] composite of the second to the seventh arrows in the ﬁrst display of the present proof. Thus, Ξ may be understood as the result of composing Ψ with the operation of taking the logarithmic derivative with respect to t and then truncating the terms of degree ≥ 2g Y n . Observe that so far we have not applied the assumption that K is a p-adic local ﬁeld. Now we proceed to apply this assumption. Recall from Proposition 2.8, (i), (ii); Proposition 2.11, (ii) [cf. also the initial portions of Proposition 2.6 and Deﬁnition 2.7], that rank Z p T qtr,n = rank Z p T tor,n = p n (l m − l m−1 )e X ; 2g Y n = 2(p n (g Y 0 − 1) + 1), where e X ≥ 1. Thus, one veriﬁes immediately that there exists a real number C g Y ,d K that depends only on g Y = g Y 0 and d K = [K : Q p ] such that for any positive integer n ≥ C g Y ,d K , it holds that p n (l m − l m−1 )e X > d K (log p (2(p n (g Y 0 − 1) + 1)) + 1). 69 In particular, we conclude from Lemma 2.15 that there exists a homomorphism m ) such that Ξ(f ) = 0, and ord(Ξ(f )) + 1 is not a [non- f ∈ Hom O K ( T qtr,n , G negative integral] power of p. Fix such a homomorphism f . Then note that it follows from our choice of f that we may write a i t i ∈ O K [[t]] × Ψ(f ) = 1 + i≥1 [cf. Proposition 2.13, (ii)], where, if we write i 0 for the smallest positive integer i such that a i = 0, then i 0 is not a [nonnegative integral] power of p. In the following, we shall apply Proposition 1.6, where we take “g(t)” to be Ψ(f ) and apply the isomorphism of topological O K -algebras ∼ Y ,y O K [[t]] → O n n def determined by t, to complete the proof of Theorem 2.16. Write N = μ + 1, where μ is the “μ” that results from applying Proposition 1.6, (i); ψ η : W n −→ Y n for the [connected] ﬁnite étale Galois covering over K obtained by pulling-back the morphism induced by multiplication by p N on A n via the composite mor- phism Y n → U y n → J n A n [cf. Proposition 2.12]. Next, let us observe that it follows immediately from the various deﬁnitions of the morphisms involved that the composite morphism φ c ◦ λ h ◦ τ ◦ φ c | U : U c → Spec O K [[t]] 1 2 c 2 2 [cf. the two commutative diagrams of Proposition 1.6, (i)], together with the ∼ isomorphism O K [[t]] → O Y n ,y n , allow one to regard the p-adic completion R Y of Γ(O U c 2 , U c 2 ) at the generic point of (U c 2 ) s as the p-adic completion of the ring of integers R Y of a certain discrete residue-transcendental p-valuation on the function ﬁeld of Y n . Write R W/Y for the normalization of R Y in the function ﬁeld of W n . Thus, since R Y is [a localization of a ring of ﬁnite type over the complete discrete valuation ring O K , hence] excellent, it follows that R W/Y is ﬁnite over R Y . Next, let us observe that it follows from the relations ι ◦ λ g ◦ φ c ◦ λ h = (p μ ) ◦ ξ g , 1 ξ g ◦ τ ◦ (φ c 2 | U c 2 ) = ι ◦ (φ π p | U πp ) ◦ θ g , ι ◦ (φ π p | U πp ) ◦ θ g ◦ f Y = (p) ◦ ι ◦ (φ π | U π ) ◦ θ Y in the ﬁrst and second commutative diagrams of Proposition 1.6, (i), and the commutative diagram of Remark 1.6.1 that we obtain relations ι ◦ λ g ◦ φ c ◦ λ h ◦ τ ◦ (φ c | U ) ◦ f Y = (p μ ) ◦ ξ g ◦ τ ◦ (φ c | U ) ◦ f Y 1 2 c 2 2 c 2 = (p ) ◦ ι ◦ (φ π p | U πp ) ◦ θ g ◦ f Y μ = (p μ ) ◦ (p) ◦ ι ◦ (φ π | U π ) ◦ θ Y , where 70 • the composite of the ﬁrst and second equalities implies that, after possi- bly replacing K by a ﬁnite extension ﬁeld of K [which in fact may be Y ) K - taken to be unramiﬁed — cf. Lemma 2.10, (i)], the tautological ( R def Y ) K = R Y ⊗ O K, lifts to an valued point y R of Y n , where we write ( R K ∗ ( R Y ) K -valued point y R of a certain intermediate covering W n → Y n ∗ of W n → Y n that corresponds to multiplication by p μ on the codomain of m ), while the homomorphism f ∈ Hom O K ( T qtr,n , G • the third equality implies [cf. also the “essentially cartesian” nature of ∗ the squares in the commutative diagram of Remark 1.6.1] that y R lifts def W ) K = R W ⊗ O K, W ) K -valued point w R of W n , where ( R to an ( R K and R W denotes the p-adic completion of some localization R W of R W/Y W admits a at a maximal ideal of R W/Y such that the spectrum of R tautological isomorphism over R Y to the spectrum of the p-adic completion of “Γ(O Y , Y )” at the generic point of “Y s ” [i.e., where the quotation marks refer to the notation of Proposition 1.6, (ii)]. In particular, since i 0 is not a [nonnegative integral] power of p, it follows from W , hence also Proposition 1.6, (ii), and Remark 1.6.2 that the residue ﬁeld of R the residue ﬁeld of R W , is the function ﬁeld of a curve over the residue ﬁeld of O K of genus ≥ 1. Thus, we conclude from Proposition 2.3, (ii), (iii), that, after possibly replacing K by a ﬁnite extension ﬁeld of K, there exist a compactiﬁed semistable model W n of W n over O K , together with a dominant morphism ψ : W n −→ Y n over O K , such that • ψ restricts to the ﬁnite étale Galois covering ψ η : W n → Y n ; • R W is the local ring of W n at the generic point of an irreducible component D of (W n ) s whose normalization is of genus ≥ 1; • ψ(D) = (y n ) s ∈ (Y n ) s . This completes the proof of Theorem 2.16. Theorem 2.17 (Resolution of nonsingularities for arbitrary hyperbolic curves over p-adic local ﬁelds). Let Σ ⊆ Primes be a subset of cardinality ≥ 2; K a p-adic local ﬁeld, for some p ∈ Σ; X a hyperbolic curve over K; L a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p that contains K as a topological subﬁeld. Then X L satisﬁes Σ-RNS if and only if the residue ﬁeld of L is algebraic over the ﬁnite ﬁeld of cardinality p. Proof. First, we observe that it follows formally from Remark 2.2.3, (v), that it suﬃces to verify that X satisﬁes Σ-RNS. Next, we observe that it follows 71 immediately from the various deﬁnitions involved that we may assume without loss of generality that X has stable reduction over K. Write X for the [unique, up to unique isomorphism] compactiﬁed stable model of X over O K . Then, in light of Propositions 2.3, (xii); 2.4, (i), (ii), by replacing X by the [unique, up to unique isomorphism] smooth compactiﬁcation of a suitable connected geometrically pro-Σ ﬁnite étale covering of X, we may assume without loss of generality that: • X is a proper hyperbolic curve over K, • X s is split, • X s is singular, and • every irreducible component of X s is a smooth curve of genus ≥ 2. In particular, X now satisﬁes the assumptions imposed in the respective initial portions of Proposition 2.6 and Deﬁnition 2.7. Next, observe that since the covering Y n → Y is combinatorial [cf. Deﬁnition 2.7, (ii)], it follows immediately that this covering induces a surjection Y n (K) Y (K) on K-rational points. Thus, it follows immediately from Proposition 2.4, (iii), and Theorem 2.16 that X satisﬁes Σ-RNS. This completes the proof of Theorem 2.17. 3 Point-theoreticity, metric-admissibility, and arith- metic cuspidalization Let p be a prime number. In the present section, we ﬁrst recall the well- known classiﬁcation of the points of the topological Berkovich space associated to a proper hyperbolic curve over a mixed characteristic complete discrete val- uation ﬁeld via the notion of type i points, where i ∈ {1, 2, 3, 4} [cf. Deﬁ- nition 3.1]. Next, we introduce a certain combinatorial classiﬁcation of the VE-chains considered in §2 [cf. Deﬁnition 3.2] and observe that this classiﬁca- tion of VE-chains leads naturally to a purely combinatorial characterization of the well-known classiﬁcation via type i points mentioned above [cf. Proposi- tions 3.3, 3.4]. This combinatorial classiﬁcation/characterization [cf. also the approach of Propositions 3.7, 3.8] was motivated by the argument applied in the proof of [CbTpIV], Theorem A.7. We then apply the theory of §2 to give a group-theoretic characterization, motivated by [but by no means identical to] the characterization of [Lpg2], §4, of the type i points in terms of the geometric Σ-tempered fundamental group of the hyperbolic curve [cf. Propositions 3.5, 3.9]. Then, by combining this group-theoretic characterization with [AbsTopII], Corollary 2.9, we prove an absolute version of the Grothendieck Conjecture for hyperbolic curves over p-adic local ﬁelds [cf. Theorem 3.12]. This settles one of the major open questions in anabelian geometry. As a corollary of this absolute version of the Grothendieck Conjecture for hyperbolic curves over p-adic local 72 ﬁelds, together with [HMM], Theorem A, we also obtain an absolute version of the Grothendieck Conjecture for conﬁguration spaces associated to hyper- bolic curves over p-adic local ﬁelds [cf. Theorem 3.13]. We then switch gears to discuss metric-admissibility for p-adic hyperbolic curves. This discussion of metric-admissibility leads to a proof that all of the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature in fact coin- cide [cf. Theorem 3.16]. Moreover, as an application of Corollary 2.5, (i), and the theory developed in the present section, together with the theory of metric- admissibility developed in [CbTpIII], §3, we obtain a construction of a certain type of arithmetic cuspidalization of the [Primes-] tempered fundamental group of a hyperbolic curve over Q p [cf. Theorem 3.20]. Deﬁnition 3.1. Let Σ ⊆ Primes be a nonempty subset; K a mixed charac- teristic complete discrete valuation ﬁeld of residue characteristic p; X a proper → X a universal geometrically pro-Σ covering of X. hyperbolic curve over K; X Write Ω for the p-adic completion of [some ﬁxed] K; (−) an for the topological an [cf. Berkovich space associated to (−). Let x ∈ X an be an element; x̃ ∈ X Proposition 2.3, (vii), (viii)] a lifting of x. Then: (i) We shall say that x̃ is of type 1 if x̃ is determined by a point-theoretic of X associated to some point p-valuation on the function ﬁeld K( X) ∈ X(Ω) [cf. Deﬁnition 2.2, (ii)]. (ii) We shall say that x̃ is of type 2 if x̃ is determined by an inverse system of discrete residue-transcendental p-valuations associated to irreducible components of the special ﬁbers of [compactiﬁed] semistable models with split reduction of the domain curves of connected ﬁnite étale coverings → Z → X [cf. Proposition 2.3, Z → X equipped with a factorization X (ii), (iii)]. (iii) We shall say that x̃ is of type 3 if there exist a ﬁnite extension ﬁeld L of K, a [compactiﬁed] semistable model with split reduction X of X L over O L , and a node e of X s such that x̃ arises as the inverse image of a lifting tor of some element ∈ D e ⊆ VE(X ) tor [cf. Deﬁnition 2.2, (vi)] ∈ VE( X) ∼ an → tor [cf. Proposition 2.3, (viii)]. via the homeomorphism X VE( X) (iv) We shall say that x̃ is of type 4 if, for each i ∈ {1, 2, 3}, x̃ is not of type i. (v) For each i ∈ {1, 2, 3, 4}, we shall say that x is of type i if x̃ is of type i. [One veriﬁes immediately that, for each i ∈ {1, 2, 3, 4}, the condition that x is of type i is independent of the choices of Σ and x̃.] For each an [i] ⊆ X an ) i ∈ {1, 2, 3, 4}, we shall write X an [i] ⊆ X an (respectively, X an ). for the subset of points of type i of X an (respectively, X 73 Remark 3.1.1. In the notation of Deﬁnition 3.1, we observe that an = X an [1] ∪ X an [2] ∪ X an [3] ∪ X an [4]; X X an = X an [1] ∪ X an [2] ∪ X an [3] ∪ X an [4], and, moreover, for each pair of distinct i, j ∈ {1, 2, 3, 4}, an [j] = ∅; an [i] ∩ X X X an [i] ∩ X an [j] = ∅. an [j] = ∅. First, by considering an [i] ∩ X Indeed, it suﬃces to verify that X the residue ﬁelds of the valuation rings under consideration, we conclude that an [2] = ∅. Next, we observe that it follows immediately from the an [1] ∩ X X discussion of the construction of “VE(Z) tor ” in Deﬁnition 2.2, (vi), that, for an [i] ∩ X an [3] = ∅. Finally, we observe that it is a tautology each i ∈ {1, 2}, X an [i] ∩ X an [4] = ∅. This completes the proof of that, for each i ∈ {1, 2, 3}, X relations in the above display. Deﬁnition 3.2. We maintain the notation of Deﬁnition 3.1. Let c = (c Z ) Z∈S ∈ where S denotes the directed set of [compactiﬁed] semistable models VE( X), [cf. Deﬁnition 2.2, (iii)]. Write V c ⊆ S that appear in the deﬁnition of VE( X) (respectively, E c ⊆ S) for the subset of [compactiﬁed] semistable models Z such that c Z is a vertex (respectively, an edge). Then: (i) We shall say that c is asymptotically verticial (respectively, asymptotically edge-like) if the subset V c ⊆ S (respectively, E c ⊆ S) forms a coﬁnal subset of S. [In particular, if c is asymptotically verticial (respectively, asymptotically edge-like), then V c (respectively, E c ) may be regarded as a directed set in a natural way.] (ii) Suppose that c is asymptotically verticial. Then we shall say that c is strongly verticial if there exists a coﬁnal subset S c ⊆ V c satisfying the following condition: Let Z 1 , Z 2 ∈ S c be distinct elements such that Z 2 dominates Z 1 . Then the generic point of the irreducible component of (Z 2 ) s that corresponds to c Z 2 maps to the generic point of the irreducible component of (Z 1 ) s that corresponds to c Z 1 via the dominant morphism Z 2 → Z 1 . (iii) Suppose that c is asymptotically verticial. Then we shall say that c is weakly verticial if there exists a coﬁnal subset S c ⊆ V c satisfying the following condition: Let Z 1 , Z 2 ∈ S c be distinct elements such that Z 2 dominates Z 1 . Then the generic point of the irreducible component of (Z 2 ) s that corresponds to c Z 2 maps to a closed point in the interior of the irreducible component of (Z 1 ) s that corresponds to c Z 1 via the dominant morphism Z 2 → Z 1 . 74 (iv) Suppose that c is asymptotically edge-like. Then we shall say that c is weakly edge-like if there exists a coﬁnal subset S c ⊆ E c satisfying the following condition: Let Z 1 , Z 2 ∈ S c be distinct elements such that Z 2 dominates Z 1 . Then there exists a toral compactiﬁed semistable model Z 1 ∗ relative to Z 1 such that the dominant morphism Z 2 → Z 1 admits a factorization Z 2 → Z 1 ∗ → Z 1 , and the node of (Z 2 ) s that corresponds to c Z 2 maps to a closed point in the interior of an irreducible component of (Z 1 ∗ ) s [that necessarily lies over the node of (Z 1 ) s that corresponds to c Z 1 ] via the dominant morphism Z 2 → Z 1 ∗ . (v) Suppose that c is asymptotically edge-like. Then we shall say that c is strongly edge-like if there exists a coﬁnal subset S c ⊆ E c satisfying the following condition: Let Z 1 , Z 2 ∈ S c be distinct elements such that Z 2 dominates Z 1 . Then, for each toral compactiﬁed semistable model Z 1 ∗ relative to Z 1 that admits a factorization Z 2 → Z 1 ∗ → Z 1 , the node of (Z 2 ) s that corresponds to c Z 2 maps to a node of (Z 1 ∗ ) s [that necessarily lies over the node of (Z 1 ) s that corresponds to c Z 1 ] via the dominant morphism Z 2 → Z 1 ∗ . (vi) We shall write vtc ⊆ VE( X); VE( X) edg ⊆ VE( X); VE( X) str-vtc ⊆ VE( X); VE( X) wk-vtc ⊆ VE( X); VE( X) str-edg ⊆ VE( X); VE( X) wk-edg ⊆ VE( X), VE( X) respectively, for the subsets of asymptotically verticial VE-chains, asymp- totically edge-like VE-chains, strongly verticial VE-chains, weakly verti- cial VE-chains, strongly edge-like VE-chains, and weakly edge-like VE- chains. Also, for each ∈ {vtc, edg, str-vtc, wk-vtc, str-edg, wk-edg}, we shall write prim, = VE( X) prim ∩ VE( X) VE( X) def (⊆ VE( X)). (vii) Let Z ∈ S. Then in the notation of Deﬁnition 2.2, (vi), we shall write def VE(Z) tor,rat = V w (⊆ VE(Z) tor ); D e (⊆ VE(Z) tor ). w∈V(Z) def VE(Z) tor,irr = e∈E(Z) 75 Note that one veriﬁes immediately that VE(Z) tor = VE(Z) tor,rat VE(Z) tor,irr , and that, for any Z 1 , Z 2 ∈ S such that Z 2 dominates Z 1 , the natural map VE(Z 2 ) tor → VE(Z 1 ) tor induces a map VE(Z 2 ) tor,rat → VE(Z 1 ) tor,rat [cf. the discussion of Deﬁnition 2.2, (vi)]. Remark 3.2.1. In the notation of Deﬁnition 3.2, we observe that it follows im- mediately from the various deﬁnitions involved [cf. also Remarks 2.1.4, 2.1.5; Deﬁnition 2.2, (vi)] that: str-vtc ∩ VE( X) wk-vtc = ∅; VE( X) str-edg ∩ VE( X) wk-edg = ∅; VE( X) vtc = VE( X) str-vtc ∪ VE( X) wk-vtc ; VE( X) edg = VE( X) str-edg ∪ VE( X) wk-edg ; VE( X) = VE( X) vtc ∪ VE( X) edg . VE( X) Proposition 3.3 (Elementary properties of the combinatorial classiﬁ- cation of VE-chains). In the notation of Deﬁnition 3.2, by forming the in- ductive limit of the natural log structures on the [compactiﬁed] semistable models Z ∈ S, we obtain an ind-log structure on the pro-scheme lim Z∈S Z. Then the ←− following hold: (i) Let c ∈ VE( X). Write z̃ c for the center on lim Z∈S Z of the valuation ←− ring R c associated to c [cf. Proposition 2.3, (vi)]; M c pf for the perfection of the inductive limit monoid obtained by forming the stalk at z̃ c of the characteristic of the ind-log structure on the pro-scheme lim Z∈S Z. Then, ←− str-edg ), then M pf is isomorphic vtc (respectively, c ∈ VE( X) if c ∈ VE( X) c to Q ≥0 (respectively, Q ≥0 × Q ≥0 ). (ii) The following relations hold: str-vtc ∩ VE( X) str-edg = ∅; VE( X) str-vtc ∩ VE( X) wk-vtc = ∅; VE( X) wk-vtc ∩ VE( X) str-edg = ∅; VE( X) wk-edg ⊆ VE( X) wk-vtc ; VE( X) = VE( X) str-vtc ∪ VE( X) wk-vtc ∪ VE( X) str-edg . VE( X) 76 Proof. Assertion (i) follows immediately from the well-known log structure of M c pf [cf. also the discussion of the subsection in Notations and Conventions en- titled “Log schemes”], together with the various deﬁnitions involved. Assertion (ii) follows immediately from assertion (i), together with the various deﬁni- tions involved [cf. also Remark 3.2.1]. This completes the proof of Proposition 3.3. Proposition 3.4 (Characterization of points of type 2 and 3 via the combinatorial classiﬁcation of VE-chains). We maintain the notation of Deﬁnition 3.2. Then the following hold: \ VE( X) prim . Then nonprim def = VE( X) (i) Write VE( X) nonprim ∪ VE( X) prim,str-edg . str-edg = VE( X) VE( X) Moreover, the unique nontrivial generization of a nonprimitive VE-chain [cf. Proposition 2.3, (x)] is strongly verticial. (ii) For Z ∈ S, write τ X str-edg str-edg ⊆ VE( X) −→ tor −→ VE(Z) tor : VE( X) VE( X) τ X,Z for the natural composite map [cf. Proposition 2.3, (viii)]. Then for any str-edg induces a map Z ∈ S, τ X,Z nonprim → VE(Z) tor,rat VE( X) prim,str-edg , there exists an element [cf. (i)]. Moreover, for each c ∈ VE( X) Z c ∈ S such that for every Z ∈ S that dominates Z c , str-edg (c) ∈ VE(Z) tor,irr τ X,Z [cf. (i)]. Finally, Z c ∈ S may be taken to be a [compactiﬁed] semistable model with split reduction of X L over O L for some ﬁnite extension ﬁeld L of K. ∼ an prim → X [cf. Proposition 2.3, (viii)] determines (iii) The bijection VE( X) bijections ∼ an [2]; prim,str-vtc → X VE( X) ∼ prim,str-edg → X an [3]. VE( X) (iv) Let Y be a proper hyperbolic curve over K; f : Y → X a dominant def morphism over K; y ∈ Y an . Write x = f (y) ∈ X an . Then, for each i ∈ {1, 2, 3, 4}, y is of type i if and only if x is of type i. 77 Proof. Assertion (i) follows immediately from the various deﬁnitions involved. str-edg . First, suppose that Next, we verify assertion (ii). Let c ∈ VE( X) nonprim prim,str-vtc c ∈ VE( X) . Write c ∈ VE( X) for the unique nontrivial gener- ization of c [cf. assertion (i)]. Then it follows immediately from the deﬁnition of τ X in the proof of Proposition 2.3, (viii), that τ X (c) = τ X (c ). In particular, it prim,str-vtc that τ (c ) maps to follows immediately from the fact that c ∈ VE( X) X str-edg (c) ∈ VE(Z) tor,rat , an element of VE(Z) tor,rat for any Z ∈ S, hence that τ X,Z prim,str-edg . Let Z c ∈ S c for some as desired. Next, suppose that c ∈ VE( X) “S c ” as in Deﬁnition 3.2, (v). Let Z ∈ S be an element that dominates Z c . Then one veriﬁes immediately [cf. also the ﬁnal portion of Deﬁnition 3.2, (vii)] str-edg (c) ∈ VE(Z) tor,rat implies a contradiction to the condi- that any relation τ X str-edg tion of Deﬁnition 3.2, (v). Thus, we conclude that τ X,Z (c) ∈ VE(Z) tor,irr , as desired. The fact that Z c ∈ S may be taken to be a [compactiﬁed] semistable model with split reduction of X L over O L for some ﬁnite extension ﬁeld L of K follows immediately from Proposition 2.3, (iii), (iv). This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii), together with the various deﬁnitions involved [cf. also the ﬁnal portion of Deﬁnition 3.2, (vii); Proposition 3.3, (ii)]. Finally, we consider assertion (iv). First, we observe that the asserted equivalence follows immediately from the various deﬁnitions involved [cf. also Proposition 2.3, (ii), (iii), in the case where i = 2] when i ∈ {1, 2}. Thus, it suﬃces to verify the asserted equivalence when i = 3. When i = 3, suﬃciency follows immediately, in light of assertion (iii) and Proposition 3.3, (ii), from Deﬁnition 3.1, (iii) [cf. also the discussion of Deﬁnition 2.2, (vi)]. On the other hand, it follows immediately, by replacing Y by the normalization of Y in the Galois closure of the ﬁnite extension of function ﬁelds determined by f and applying the suﬃciency that has already been veriﬁed, that to verify necessity when i = 3, we may assume without loss of generality that the ﬁnite extension of function ﬁelds determined by f is Galois. But then the desired necessity follows immediately from the ﬁnal portion of assertion (ii), together with Proposition 2.3, (iii), (iv) [cf. also the discussion of Deﬁnition 2.2, (vi)]. This completes the proof of Proposition 3.4. Proposition 3.5 (Types of points and geometrically pro-l decomposi- tion groups). In the notation of Deﬁnition 3.2, let l ∈ Σ \ {p}. Suppose that X satisﬁes Σ-RNS. Then the following hold: (i) Let wk-vtc ; c ∈ VE( X) str-edg ). str-vtc (respectively, c ∈ VE( X) c ∈ VE( X) →) Z → For each connected geometrically pro-Σ ﬁnite étale covering ( X def {l} X, write D Z,c ⊆ Δ lZ = Δ Z for the decomposition subgroup associated to c of the geometric pro-l fundamental group of Z [cf. the subsection in 78 Notations and Conventions entitled “Fundamental groups”]. Then there →) Y → X exists a connected geometrically pro-Σ ﬁnite étale covering ( X of X such that, for each connected geometrically pro-Σ ﬁnite étale covering →) Z → Y of Y , D Z,c is isomorphic to a [nonabelian] pro-l surface ( X group [cf. [MT], Deﬁnition 1.2] (respectively, the trivial group; Z l ). an and the subset X an [i] ⊆ X an may (ii) Let i ∈ {2, 3}. Then the set X be reconstructed, functorially with respect to isomorphisms of topological groups, from the underlying topological group of the geometric Σ-tempered fundamental group of X [cf. the subsection in Notations and Conventions entitled “Fundamental groups”]. (iii) Let Y , Z be [not necessarily proper!] hyperbolic curves over K that satisfy Σ-RNS; Y → Y , Z → Z universal geometrically pro-Σ coverings of Y , Z, respectively; f : Y → Z a dominant morphism over K; H ⊆ G K a closed subgroup such that the restriction to H of the l-adic cyclotomic charac- ter of K has open image, and, moreover, the intersection H ∩I K of H with the inertia subgroup I K of G K admits a surjection to [the proﬁnite group] (Σ) def (Σ) def = Gal( Y /Y ), s Z : H → Π = Gal( X/X) sections Z l ; s Y : H → Π Y Z (Σ) of the restrictions to H of the respective natural surjections Π Y G K , (Σ) (Σ) Π Z G K such that s Y is mapped, up to Π Z -conjugation, by f to s Z via the map induced by f on geometrically pro-Σ fundamental groups. Write Ω H ⊆ Ω for the subﬁeld of Ω ﬁxed by H. Then s Y arises from a(n) [necessarily unique] Ω H -rational point ∈ Y (Ω H ) if and only if s Z arises from a(n) [necessarily unique] Ω H -rational point ∈ Z(Ω H ). Proof. Since X satisﬁes Σ-RNS [cf. Deﬁnition 2.2, (vii)], assertion (i) follows immediately from the well-known structure of the maximal pro-l quotient of the admissible fundamental group of a stable curve over a separably closed ﬁeld of characteristic p [cf. e.g., [SemiAn], Example 2.10], together with the various def- initions involved. Assertion (ii) follows immediately from assertion (i), together with Corollary 2.5, (i); Proposition 3.4, (iii) [cf. also Proposition 3.3, (ii)]. Fi- nally, we consider assertion (iii). First, we observe that it follows immediately (Σ) from the proﬁnite nature of the topological group Π Y that, by replacing Y and Z by the smooth compactiﬁcations of the various ﬁnite étale coverings of Y and Z corresponding, respectively, to suitable open neighborhoods of the image of (Σ) (Σ) s Y in Π Y × G K H and the image of s Z in Π Z × G K H, we may assume without loss of generality that Y and Z are proper. Next, we observe that the various uniqueness assertions in the statement of Proposition 3.5, (iii), follow immedi- ately from the ﬁnal portion of Proposition 2.4, (vii), and that necessity follows immediately from the various deﬁnitions involved. Thus, it suﬃces to verify suﬃciency. On the other hand, suﬃciency follows, in light of the equivalences of Proposition 3.4, (iv), formally from the ﬁnal portion of Proposition 2.4, (vii). This completes the proof of Proposition 3.5. 79 Deﬁnition 3.6. Let K be a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; X a proper hyperbolic curve over K. Write K(X K ) for the function ﬁeld of X K . Let v be a p-valuation on K(X K ). Write (O K ⊆) O v ⊆ K(X K ) for the valuation ring associated to v. [Note that it follows immediately from the well-known theory of one-dimensional function def ﬁelds that (O v ) K = O v · K ⊆ K(X K ) is equal either to K(X K ) or to the discrete valuation ring associated to a closed point of X K .] Then: (i) Let M be an O v -module. Then we shall say that M is bounded if the image def of M via the natural morphism M → M K = M ⊗ O K K is contained in a ﬁnitely generated O v -submodule of M K . We shall say that M is unbounded if M is not bounded. (ii) We shall say that the p-valuation v is diﬀerentially bounded (respectively, diﬀerentially unbounded) if the O v -module of relative diﬀerentials Ω O v /O K is bounded (respectively, unbounded). Proposition 3.7 (Approximation of closed points of the generic ﬁber via generic points of special ﬁbers). Let K be a mixed characteristic com- plete discrete valuation ﬁeld of residue characteristic p; K = K 0 ⊆ K 1 ⊆ · · · ⊆ K i ⊆ · · · an ascending chain of ﬁnite extension ﬁelds of K contained in K and indexed by N. Write Ω for the p-adic completion of K. Let X be a hyperbolic curve over K. For each i ∈ N, let X i be a compactiﬁed semistable model with split reduction of X K i over O K i ; φ i+1 : X i+1 −→ X i × O Ki O K i+1 a dominant morphism over O K i+1 that induces the identity automorphism on the generic ﬁber; v i an irreducible component of (X i ) s . Suppose that, for each i ∈ N, the projection to X i of φ i+1 (v i+1 ) is a closed point x i ∈ (X i ) s ⊆ X i of (X i ) s that lies in the smooth locus of v i . Then the following hold: (i) For each i ∈ N, let ψ i : Spec O K i → X i be a section whose image contains def x i . Then there exists a collection {t i , γ i+1 , π i+1 } i∈N of elements t i ∈ A i = O X i ,x i and γ i+1 , π i+1 ∈ m K i+1 such that, for each i ∈ N, t i ∈ A i is a generator of the ideal that deﬁnes the scheme-theoretic image of ψ i , and t i = γ i+1 + π i+1 t i+1 , where we regard A i as a subring of A i+1 via the injection A i → A i+1 induced by the composite X i+1 → X i [which maps x i+1 → x i ] of φ i+1 with the projection to X i . 80 def (ii) We maintain the notation of (i). For each positive integer i, write l i = v p (π i ). Suppose that the equality l i = +∞ i≥1 holds. Then there exists a closed point x Ω of X Ω such that, for each i ∈ N, the center on X i of the closed point x Ω of X Ω , hence also of the point-theoretic p-valuation on the function ﬁeld of X K i determined by the closed point x Ω of X Ω , coincides with x i . Proof. First, we verify assertion (i). We construct elements t i ∈ A i and γ i+1 , π i+1 ∈ m K i+1 , for i ∈ N, by induction on i ∈ N. Let t 0 ∈ A 0 be a generator of the ideal that deﬁnes the scheme-theoretic image of ψ 0 ; i ∈ N. Suppose that the elements t j+1 , γ j+1 , and π j+1 have been constructed for j ∈ N such that j < i. Write γ i+1 ∈ m K i+1 for the image of t i via the composite homomor- phism A i ⊆ A i+1 → O K i+1 induced by ψ i+1 . Next, observe that the image in A i+1 ⊗ O Ki+1 K i+1 of t i − γ i+1 is a generator of the maximal ideal associated to the closed point of X K i+1 determined by ψ i+1 . Thus, since x i+1 lies in the smooth locus of (X i+1 ) s , and A i+1 is a regular local ring, hence a unique fac- torization domain, we conclude that there exists an element π i+1 ∈ m K i+1 such that t i −γ i+1 = π i+1 t i+1 for some generator t i+1 ∈ A i+1 of the ideal that deﬁnes the scheme-theoretic image of ψ i+1 . This completes the proof of assertion (i). Next, we verify assertion (ii). Suppose that i≥1 l i = +∞. For each i ∈ N, ﬁx elements t i ∈ A i and γ i+1 , π i+1 ∈ m K i+1 as in the statement of assertion (i). For each positive integer j, write def s j = γ j · π i . 1≤i≤j−1 it follows immediately from our assump- Then since O Ω is p-adically complete, tion that i≥1 l i = +∞ that j≥1 s j converges to an element γ ∈ m Ω . Write x Ω for the closed point of X Ω determined by the homomorphism ψ Ω : A 0 → O Ω over O K 0 that maps t 0 → γ. Now observe that it follows immediately from the deﬁnition of γ that, for each i ∈ N, ψ Ω extends uniquely to a homomorphism A i → O Ω over O K i . On the other hand, the existence of such unique exten- sions implies that the center on X i of the closed point x Ω of X Ω , hence also of the point-theoretic p-valuation on the function ﬁeld of X K i determined by the closed point x Ω of X Ω , coincides with x i . This completes the proof of assertion (ii), hence of Proposition 3.7. Proposition 3.8 (Characterization of points of type 1 via diﬀerentially unboundedness). In the notation of Deﬁnition 3.6 [cf. also the notation of (⊇ K(X )) that restricts Proposition 2.3, (viii)], let v be a p-valuation of K( X) K to v on K(X K ) [cf. Remark 2.2.4]. Suppose that v is primitive. Then the point an associated to v [cf. Proposition 2.3, (viii)] is of type 1 if and only if x v ∈ X v is diﬀerentially unbounded. 81 an . Note that, in Proof. Write x v ∈ X an for the point determined by x v ∈ X light of Remark 3.1.1, it suﬃces to verify that if x v is of type 1 (respectively, of type i ∈ {2, 3, 4}), then v is diﬀerentially unbounded (respectively, diﬀerentially bounded). First, we verify that if x v is of type 1, then v is diﬀerentially unbounded. Suppose that x v is of type 1. Write Ω for the p-adic completion of K; e v : O v → O Ω for the natural evaluation homomorphism over O K associated to x v . Next, observe that, since O v ⊗ O K K is a valuation ring [contained in the function ﬁeld K(X K ) of the hyperbolic curve X K over K] that contains K, and whose ﬁeld of fractions coincides with K(X K ) [cf. Deﬁnition 2.2, (ii)], it follows immediately from the well-known theory of one-dimensional function ﬁelds over an algebraically closed ﬁeld that Ω O v /O K ⊗ O K K is a rank one free module over O v ⊗ O K K. In particular, there exists an element t ∈ O v such that e v (t) ∈ m Ω , and dt is a free generator of Ω O v /O K ⊗ O K K over O v ⊗ O K K. Next, we observe that, for any positive integer N , there exists an element a N ∈ m K such that e v (t − a N ) = e v (t) − a N ∈ p N O Ω . N ∈ O v . On the other hand, it Note that the above equation implies that t−a p N follows immediately from the deﬁnition of the module of relative diﬀerentials that N dt = d(t − a N ) = p N · d( t−a ) ∈ Ω O v /O K . p N In particular, we conclude that dt is a nonzero p-divisible element of Ω O v /O K . Thus, since dt is a free generator of Ω O v /O K ⊗ O K K over O v ⊗ O K K, the dif- ferential boundedness of v would imply that arbitrary negative integral powers of p are contained in O v , i.e., in contradiction to our assumption that v is a p-valuation. Hence we conclude that v is diﬀerentially unbounded, as desired. Next, we verify that, if x v is of type 2, then v is diﬀerentially bounded. Suppose that x v is of type 2. Then it follows immediately from Proposition 3.4, (iii), that there exist a ﬁnite extension ﬁeld L of K, a compactiﬁed semistable ∼ model X of X L over O L , and a generic point x of X s such that O v → O X ,x ⊗ O L O K . In particular, to verify that v is diﬀerentially bounded, it suﬃces to verify that Ω O X ,x /O L is a ﬁnitely generated O X ,x -module. On the other hand, this follows immediately from the fact that O X ,x is essentially of ﬁnite type over O L . Next, we verify that, if x v is of type 3, then v is diﬀerentially bounded. Suppose that x v is of type 3. Then it follows from Proposition 3.4, (iii), that the VE-chain associated to the p-valuation v is strongly edge-like. Thus, it follows immediately from Proposition 2.3, (iii), (iv) [cf. also Deﬁnition 3.2, (v), [the ﬁnal portion of] (vii); Proposition 3.4, (ii)] that there exist • an ascending chain K = K 0 ⊆ K 1 ⊆ · · · ⊆ K i ⊆ · · · of ﬁnite extension ﬁelds of K contained in K and indexed by N 82 and, for each i ∈ N, • a compactiﬁed semistable model X i with split reduction of X K i over O K i , • a dominant morphism φ i+1 : X i+1 −→ X i × O Ki O K i+1 over O K i+1 that induces the identity automorphism on the generic ﬁber, • a node e i of (X i ) s satisfying the following conditions: • For each i ∈ N, the projection to X i of φ i+1 (e i+1 ) coincides with the node e i ∈ (X i ) s ⊆ X i . • The equality O v = lim A i ⊗ O Ki O K −→ i∈N def — where, for each i ∈ N, we write A i = O X i ,e i ; the transition map is the homomorphism A i ⊗ O Ki O K → A i+1 ⊗ O Ki+1 O K induced by the composite X i+1 → X i [which maps e i+1 → e i ] of φ i+1 with the projection to X i — holds [cf. Proposition 2.3, (vi)]. def For each i ∈ N, write S i = Spec O K i ; S i log for the log scheme determined by the log structure on S i associated to the closed point of S i ; X i log for the log scheme over S i log determined by the natural log structure on X i [i.e., the multiplicative monoid of sections of O X i that are invertible on the open subscheme of X i determined by X K i ]; ω X log /S log ,e i for the stalk at e i of the sheaf of relative i i logarithmic diﬀerentials associated to the proper, log smooth morphism X i log → S i log [cf. the subsection in Notations and Conventions entitled “Log schemes”]. Then it follows immediately from the deﬁnitions of the various log structures involved that the morphism φ i+1 : X i+1 → X i × O Ki O K i+1 = X i × S i S i+1 extends to a log étale morphism of log schemes log log −→ X i log × S log S i+1 , X i+1 i which induces a natural isomorphism ∼ ω X log /S log ,e i ⊗ A i A i+1 → ω X log /S log ,e i+1 i i i+1 i+1 of A i+1 -modules, hence a natural homomorphism φ : Ω O v /O K = lim Ω A i ⊗ O K O K /O K −→ ω X log /S log ,e 0 ⊗ A 0 O v −→ 0 0 i i∈N 83 of O v -modules. Here, we note that φ induces a natural isomorphism ∼ Ω O v /O K ⊗ O K K → ω X log /S log ,e 0 ⊗ A 0 O v ⊗ O K K 0 0 of free O v ⊗ O K K-modules of rank 1. Thus, since ω X log /S log ,e 0 is a ﬁnitely 0 0 generated A 0 -module, we conclude that v is diﬀerentially bounded, as desired. Finally, we verify that, if x v is of type 4, then v is diﬀerentially bounded. Suppose that x v is of type 4. Then it follows from Proposition 3.3, (ii); Propo- sition 3.4, (iii) [cf. also Remark 3.1.1], that the VE-chain associated to the p-valuation v [cf. Proposition 2.3, (viii)] is weakly verticial. Thus, it follows immediately from Proposition 2.3, (iii), (iv) [cf. also Deﬁnition 3.2, (iii); the ﬁnal portion of Remark 2.1.4] that there exist • an ascending chain K = K 0 ⊆ K 1 ⊆ · · · ⊆ K i ⊆ · · · of ﬁnite extension ﬁelds of K contained in K and indexed by N and, for each i ∈ N, • a compactiﬁed semistable model X i with split reduction of X K i over O K i , • a dominant morphism φ i+1 : X i+1 −→ X i × O Ki O K i+1 over O K i+1 that induces the identity automorphism on the generic ﬁber, • an irreducible component v i of (X i ) s satisfying the following conditions: • For each i ∈ N, the projection to X i of φ i+1 (v i+1 ) is a closed point x i ∈ (X i ) s ⊆ X i of (X i ) s that lies in the smooth locus of v i . • For each i ∈ N, there exists a section ψ i : Spec O K i → X i whose image contains x i . • The equality O v = lim A i ⊗ O Ki O K −→ i∈N def — where, for each i ∈ N, we write A i = O X i ,x i ; the transition map is the homomorphism A i ⊗ O Ki O K → A i+1 ⊗ O Ki+1 O K induced by the composite X i+1 → X i [which maps x i+1 → x i ] of φ i+1 with the projection to X i — holds [cf. Proposition 2.3, (vi)]. 84 Thus, we are in the situation of Proposition 3.7, (i). In particular, there exists a collection {t i , γ i+1 , π i+1 } i∈N of elements t i ∈ A i and γ i+1 , π i+1 ∈ m K i+1 as in Proposition 3.7, (i), such that t i = γ i+1 + π i+1 t i+1 . Next, observe that since x i lies in the smooth locus of (X i ) s , it follows that Ω A i /O Ki is a free A i -module of rank 1 generated by dt i . In particular, since t i = γ i+1 + π i+1 t i+1 , we conclude that 1 Ω A i+1 /O Ki+1 = · Ω A i /O Ki ⊗ A i A i+1 . π i+1 Thus, it follows immediately from the equality O v = lim i∈N A i ⊗ O Ki O K that −→ 1 · Ω A 0 /O K 0 ⊗ A 0 O v . Ω O v /O K = lim −→ 1≤j≤i+1 π j i∈N Now suppose that v is diﬀerentially unbounded. For each positive integer i, def write l i = v p (π i ). Then since v is diﬀerentially unbounded, we conclude that the equality l i = +∞ i≥1 holds. Next, we observe that it follows immediately from Proposition 3.7, (ii) [cf. also the equality O v = lim i∈N A i ⊗ O Ki O K ], that v determines a closed −→ point x Ω of X Ω such that the valuation ring of the point-theoretic p-valuation on K(X K ) associated to x Ω dominates O v , hence coincides with O v . Thus, we conclude that x v is the point ∈ X an determined by x Ω , hence that x v is of type 1, in contradiction to our assumption that x v is of type 4. This completes the proof of Proposition 3.8. Proposition 3.9 (Characterization of points of type 1 via geometric Σ-tempered decomposition groups). In the notation of Deﬁnition 3.2, sup- pose that p ∈ Σ. Let x ∈ X an be an element; l ∈ Σ \ {p}; D x a decomposition of X associated to group in the geometric Σ-tempered fundamental group Δ Σ-tp X x. Then the following hold: (i) Suppose that x is of type 1. Then D x is trivial. (ii) Suppose that x is of type 4. Then there exists an open subgroup of D x that admits a continuous surjective homomorphism to Z p . In particular, D x is nontrivial. (iii) Suppose that x is of type i ∈ {2, 3}, and that X satisﬁes Σ-RNS. Then there exists an open subgroup of D x that admits a continuous surjective homomorphism to Z l . In particular, D x is nontrivial. (iv) Suppose that X satisﬁes Σ-RNS. Then x is of type 1 if and only if D x is trivial. 85 an be a lifting of x. First, we observe that assertion (i) follows Proof. Let x ∈ X immediately from the various deﬁnitions involved. Next, we verify assertion (iii). Suppose that x [or, equivalently, x ] is of type i ∈ {2, 3}, and that X satisﬁes Σ-RNS. Then it follows immediately from Proposition 3.4, (iii); Proposition 3.5, (i), that there exists an open subgroup of D x that admits a continuous surjective homomorphism to Z l . This completes the proof of assertion (iii). Assertion (iv) follows immediately from assertions (i), (ii), (iii) [cf. also Remark 3.1.1]. Thus, to complete the proof of Proposition 3.9, it suﬃces to verify assertion (ii). To verify assertion (ii), by replacing K by a suitable extension ﬁeld of K contained in Ω, we may assume without loss of generality [cf. Proposition 3.4, (iii)] that the residue ﬁeld of K is separably closed. Suppose that x [or, equivalently, x ] is of type 4, and that no open subgroup of D x admits a continuous surjective homomorphism to Z p . Write v for the primitive p-valuation on K( X) associated to x [cf. Proposition 2.3, (viii)]; v for the p-valuation obtained by restricting v to K(X K ). Then since x is of type 4, it follows from Proposition 3.8 that Ω O v /O K is bounded. In particular, by replacing K by a ﬁnite extension ﬁeld of K, if necessary, we observe that there exist • a positive integer N and • a compactiﬁed semistable model with split reduction X of X over O K such that the center z on X of the VE-chain associated to v lies in the smooth locus of X s ⊆ X , arises from a point of X valued in the residue ﬁeld of O K , and satisﬁes the following condition [cf. the portion of the proof of Proposition 3.8 concerning points of type 4]: (Ω O X ,z /O K ⊆) Ω O X ,z /O K ⊗ O X ,z O v ⊆ Ω O v /O K ⊆ 1 · Ω O X ,z /O K ⊗ O X ,z O v p N — where by a slight abuse of notation, we use the notation “⊆” to denote the various natural inclusions, and we note that since Ω O X ,z /O K is a free O X ,z - module of rank 1, the O v -module Ω O X ,z /O K ⊗ O X ,z O v is a free O v -module of rank 1. In particular, it follows immediately that the second and third inclusions of the above display induce the injections on the respective p-adic completions [cf. the discussion of Remark 2.2.4]. In the remainder of the proof of assertion (ii), we suppose that we are in the situation of Proposition 2.12. Moreover, by replacing X by a suitable geo- metrically pro-Σ connected ﬁnite étale covering of X [cf. Proposition 2.3, (xii); Deﬁnition 2.7], we may assume without loss of generality that X = Y n , X = Y n , z = (y n ) s , X ,z = O Y ,y , O n n X ,z denotes the completion of the local ring O X ,z . Write where O m ) −→ O × Ψ : Hom O K ( T qtr,n , G X ,z m ) for the assignment discussed in Proposition 2.13, (ii). Let f ∈ Hom O K ( T qtr,n , G be a nontrivial element [which exists by Propositions 2.11, (i); 2.13, (i)]. Thus, 86 the logarithmic diﬀerential def def ) θ = dΨ(f X ,z /O K = lim Ω (O X ,z /m m Ψ(f ) ∈ Ω O z )/O K ←− m≥1 — where m z denotes the maximal ideal of O X ,z ; m ranges over the positive × is = 0 [cf. the ﬁrst display, as well as the discussion integers — of Ψ(f ) ∈ O X ,z following this ﬁrst display, in the proof of Theorem 2.16, where we observe that this portion of the proof of Theorem 2.16 may be applied even in the case of the “K” — i.e., with separably closed residue ﬁeld — of the present discussion]. v for the ﬁeld of fractions of v for the p-adic completion of O v ; K Next, write O v . Since v is a real valuation [cf. Proposition 2.3, (vii); Remark 3.1.1], it follows O immediately from the ﬁnal portion of Remark 2.2.4 that the henselization of O v v . Write H ⊆ Δ Σ-tp for the closed subgroup may be regarded as a subring of O X obtained by forming the intersection of the kernels of the continuous surjective H , K( X) D x ⊆ K( X) for the subﬁelds ﬁxed homomorphisms Δ Σ-tp Z p ; K( X) X by H and D x , respectively. Then since there does not exist any continuous surjective homomorphism D x Z p , we thus conclude that D x ⊆ H, hence that H ⊆ K( X) D x ⊆ K v . K( X) On the other hand, it follows immediately from the deﬁnition of the center z on X that there exists a natural homomorphism φ : O X ,z → O v of local rings, X ,z → O v of topological local rings. which thus induces a homomorphism φ : O def = 0. Now we claim that φ is injective. Indeed, suppose that p = Ker( φ) Then since O X ,z is a regular local ring of dimension 2, and O v is p-torsion-free, it follows that p is a prime ideal of height 1 such that p ∩ O K = {0}. Next, X ,z /p)⊗ O (O K /m K ) is ﬁnite over O K /m K . Thus, since O X ,z /p observe that ( O K is a complete O K -module, we conclude that O X ,z /p is ﬁnite over O K . On the v is other hand, observe that the composite homomorphism O X ,z → O v → O X ,z /p is injective. injective, hence that the natural homomorphism O X ,z → O In particular, we conclude that O X ,z ⊗ O K K embeds into a ﬁnite dimensional K-vector space, a contradiction. This completes the proof of our claim that φ is injective. Next, we observe that since the image of Ψ(f ) is p-divisible in the multi- X ,z ⊗ O H } × [cf. Proposition 2.13, (ii)], the image plicative group { O K( X) X ,z × , hence also in the multiplicative group of Ψ(f ) in the multiplicative group K v × O v , is p-divisible. Write Ω O v /O def K = lim Ω (O v /p m ·O v )/O K , ←− m≥1 where m ranges over the positive integers. Then it follows immediately from well-known basic facts concerning modules of diﬀerentials, together with the fact that Ω O X ,z /O K is a free O X ,z -module of rank 1, that Ω O X ,z /O K = lim Ω O X ,z /O K ⊗ O X ,z O X ,z /m m z = Ω O X ,z /O K ⊗ O X ,z O X ,z ; ←− m≥1 87 Ω O v /O K = lim Ω O v /O K ⊗ Z Z/p m Z. ←− m≥1 In particular, since φ is injective, it thus follows from the discussion of the ﬁnal portion of the second paragraph of the present proof that X ,z ⊆ Ω O /O ⊗ O O v ⊆ Ω Ω O X ,z /O K = Ω O X ,z /O K ⊗ O X ,z O X ,z X ,z K O v /O , K hence that the image θ v ∈ Ω O v /O of θ in Ω O v /O is = 0. On the other hand, K K × is p-divisible, and Ω since the image of Ψ(f ) in O is, by deﬁnition, p- O v /O K v adically separated, we conclude that θ v = 0, a contradiction. This completes the proof of assertion (ii), hence of Proposition 3.9. Corollary 3.10 (Reconstruction of points of type 1 via geometric tem- pered fundamental groups). Let Σ ⊆ Primes be a subset of cardinality ≥ 2 that contains p; K a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; X a hyperbolic curve over K. Write Ω for the p-adic completion of K; Π tp (−) for the Σ-tempered fundamental group of (−); X → X tp for the universal pro-Σ covering corresponding to Π [so Gal( X/X) may be X identiﬁed with the pro-Σ completion of Π tp X ]. Suppose that X satisﬁes Σ-RNS. Then the set X(Ω) equipped with its natural action by Gal( X/X) — hence also, the quotient set X(Ω) X(Ω) — by passing to the set of Gal( X/X)-orbits, may be reconstructed, in a purely combinatorial/group-theoretic way and func- torially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp X . Proof. Recall that, for any hyperbolic curve Y over K, the set of cuspidal inertia subgroups of Π tp Y , hence also the genus of Y , may be reconstructed, in a purely combinatorial/group-theoretic way and functorially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp Y [cf. the generalized version of [SemiAn], Corollary 3.11, discussed in [AbsTopII], Remark 2.11.1, (i)]. On the other hand, in the case where X is a proper hyperbolic curve over K, we observe that Corollary 3.10 follows immediately from Proposition 3.9, (iv), and [the proof of] Corollary 2.5, (i). Thus, by applying this observation to the Σ-tempered fundamental groups of the smooth compactiﬁcations of the various [connected] geometrically pro-Σ ﬁnite étale Galois coverings of X over K of genus ≥ 2, we conclude that X(Ω) equipped with its natural action by Gal( X/X) may be reconstructed, in a purely combinatorial/group-theoretic way and functorially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp X . This completes the proof of Corollary 3.10. 88 Theorem 3.11 (Preservation of decomposition subgroups associated to closed points). For ∈ {†, ‡}, let p be a prime number; Σ ⊆ Primes a subset that contains p ; l ∈ (Σ † \{p † }) ∩ (Σ ‡ \{p ‡ }); K a mixed characteristic complete discrete valuation ﬁelds of residue characteristic p ; X a hyperbolic curve over K ; L ⊆ K a tamely ramiﬁed [not necessarily ﬁnite!] Galois extension of K that may be written as a union of ﬁnite tamely ramiﬁed Galois extensions of K in K of ramiﬁcation index prime to l. Let σ :Π (Σ † ) ∼ (Σ ‡ ) → Π ‡ † X † X ‡ L L be an isomorphism of proﬁnite groups between the geometrically pro-Σ † étale fundamental group of X L † † and the geometrically pro-Σ ‡ étale fundamental group of X L ‡ † . For ∈ {†, ‡}, write Δ Σ for the geometric pro-Σ étale fundamental X L group of X L , I L ⊆ G L for the inertia subgroup of G L , and k L for the residue ﬁeld of L . Then the following hold: (i) We have an equality p † = p ‡ , and σ induces isomorphisms of proﬁnite † ‡ ∼ ∼ → Δ Σ , G L † → G L ‡ . In particular, Σ † = Σ ‡ . Finally, if, groups Δ Σ X † X ‡ L † L ‡ for each ∈ {†, ‡}, every pro-l closed subgroup of the kernel of the l-adic cyclotomic character on G k L is trivial [cf. Remark 3.11.1 below], then, for all suﬃciently small open subgroups J † ⊆ G L † , J ‡ ⊆ G L ‡ such that ∼ σ induces an isomorphism J † → J ‡ , σ also induces an isomorphism of proﬁnite groups between the respective images of J † ∩ I L † , J ‡ ∩ I L ‡ in the maximal pro-l quotients J † (J † ) {l} , J ‡ (J ‡ ) {l} . (ii) Suppose that, for all suﬃciently small open subgroups J † ⊆ G L † , J ‡ ⊆ ∼ G L ‡ such that σ induces an isomorphism J † → J ‡ , σ also induces an isomorphism of proﬁnite groups between the respective images of J † ∩ I L † , J ‡ ∩ I L ‡ in the maximal pro-l quotients J † (J † ) {l} , J ‡ (J ‡ ) {l} . def Write Σ = Σ † = Σ ‡ [cf. (i)]. Suppose, moreover, that X † and X ‡ satisfy Σ-RNS. Then σ induces a bijection between the respective sets of decomposition subgroups associated to closed points of X L † † and X L ‡ ‡ , where ‡ denote the respective completions of L † , L ‡ . † , L L def def Proof. First, we verify assertion (i). Write τ † = σ −1 , τ ‡ = σ. For ∈ {†, ‡}, write for the unique element of {†, ‡} \ {}. Then observe that it follows immediately, by applying to τ (Δ Σ ), for both = † and = ‡, X L • the argument of the proof of [MiTs1], Corollary 4.6 [in the case where the extension L /K is ﬁnite; here, we note that in this case, it follows from (Σ ) [MiSaTs], Theorem 3.8, that if Π X is topologically ﬁnitely generated, L then the extension L /K is also ﬁnite], and 89 • [MiSaTs], Theorem 3.8 [in the case where the extension L /K is inﬁnite], that σ induces isomorphisms of proﬁnite groups † ∼ ‡ Σ Δ Σ X † → Δ X ‡ ; ∼ G L † → G L ‡ . Thus, we conclude from [MiTs2], Theorem A, (i), together with the well-known structure of geometric fundamental groups of hyperbolic curves over ﬁelds of characteristic zero [cf., e.g., [MT], Remark 1.2.2], that p † = p ‡ , and Σ † = Σ ‡ . The ﬁnal portion of assertion (i) follows immediately from the well-known structure, for ∈ {†, ‡}, of the Galois group Gal((K ) tm /K ) over K of the maximal tamely ramiﬁed extension (K ) tm of K [under the assumption that every pro-l closed subgroup of the kernel of the l-adic cyclotomic character on G k L is trivial], which implies that, for any suﬃciently small open subgroup J ⊆ G L , the image of J ∩ I L in the maximal pro-l quotient J (J ) {l} coincides with the unique maximal abelian normal closed subgroup of (J ) {l} . This completes the proof of assertion (i). Next, we verify assertion (ii). First, we note that it follows from assertion def (i) that p = p † = p ‡ . Thus, in light of our assumption on σ [cf. also assertion (i)], it follows from [CmbGC], Corollary 2.7, (i) [applied in the case where “l” is taken to be the p of the present discussion], (iii) [applied in the case where “l” is ∼ taken to be the l of the present discussion], that the isomorphism Δ Σ → Δ Σ X † X ‡ L † L † [cf. (i)] satisﬁes the condition (b ∃ ) of [CbTpIII], Proposition 3.6. In particular, by applying [CbTpIII], Proposition 3.6, (i), we conclude that the isomorphism ∼ Δ Σ → Δ Σ arises, up to composition with an inner automorphism, from an X † X ‡ L † L ‡ isomorphism between the respective geometric Σ-tempered fundamental groups of X † and X ‡ . Thus, by replacing σ by the composite of σ with an inner automorphism arising from Δ Σ , we may assume without loss of generality X ‡ L ‡ that σ arises from an isomorphism between the respective pull-backs via the natural inclusions G L † ⊆ G K † , G L ‡ ⊆ G K ‡ of the geometrically Σ-tempered fundamental groups of X † and X ‡ . † , X ‡ for the universal geometrically pro-Σ coverings corre- Next, write X (Σ) (Σ) sponding to Π † , Π ‡ , respectively; Ω † , Ω ‡ for the p-adic completions of † ‡ X † L X ‡ L K , K , respectively. Then since σ determines an isomorphism between the respective geometric Σ-tempered fundamental groups of X † and X ‡ , it follows immediately from Corollary 3.10 that σ induces a bijection ∼ ‡ † (Ω † ) → X (Ω ‡ ) X that is compatible with the respective natural actions of Π (Σ) (Σ) , Π ‡ . X † † X † L L Thus, in light of [Tate], §3.3, Theorem 1 [which, as is easily veriﬁed, admits a routine ‡ , † , L generalization to mixed characteristic complete valuation ﬁelds such as L i.e., whose valuations are not necessarily discrete [but nonetheless tamely ram- iﬁed over some discrete valuation], and whose residue ﬁelds are not necessarily 90 perfect], we conclude that σ induces a bijection between the respective sets of decomposition subgroups associated to closed points of X L † † and X L ‡ ‡ . This completes the proof of assertion (ii), hence of Theorem 3.11. Remark 3.11.1. In passing, we observe that the condition concerning the kernel of the l-adic cyclotomic character on G k L that appears in the ﬁnal portion of Theorem 3.11, (i), is satisﬁed if k L is either separably closed or algebraic over the ﬁnite ﬁeld of cardinality p. We are now in a position to verify an absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local ﬁelds [cf. Theorem 3.12 below], which is one of the central open questions in anabelian geometry. Theorem 3.12 (Absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local ﬁelds). Let p † , p ‡ be prime numbers; Σ ⊆ Primes a subset of cardinality ≥ 2 that contains p † and p ‡ ; K † , K ‡ mixed characteristic local ﬁelds of residue characteristic p † , p ‡ , respectively; X † , X ‡ hyperbolic curves over K † , K ‡ , respectively. Then the natural map Isom(X † , X ‡ ) −→ OutIsom(Π X † , Π X ‡ ) (Σ) (Σ) is bijective. Proof. First, we observe that any isomorphism of schemes between X † and X ‡ necessarily lies over an isomorphism of ﬁelds between K † and K ‡ . [Indeed, this × follows immediately by considering subgroups of the groups of units Γ(X † , O X † ), × ‡ Γ(X , O X ‡ ) whose unions with {0} are closed under addition.] Now Theorem 3.12 follows immediately by combining Theorems 2.17; 3.11, (i), (ii) [cf. also Remark 3.11.1], of the present paper with [AbsTopII], Corollary 2.9. Remark 3.12.1. Theorem 3.12 may be regarded as a complete aﬃrmative resolu- tion of the absolute version of the Grothendieck Conjecture for hyperbolic curves over p-adic local ﬁelds in the geometrically pro-Σ case, where Σ ⊆ Primes is a subset of cardinality ≥ 2 that contains the residue characteristic of the base ﬁeld. On the other hand, the following questions remain open, to the authors’ knowledge, at the time of writing of the present paper: Question 1: Can one prove a geometrically pro-p version of the abso- lute Grothendieck Conjecture for hyperbolic curves over p-adic local ﬁelds? In this context, we observe that certain partial results in this direction are obtained in [Hgsh]. Question 2: Can one prove an absolute version of the Grothendieck Conjecture for hyperbolic curves over more general base ﬁelds? For instance, one may consider the case where the base ﬁelds are mixed 91 characteristic complete discrete valuation ﬁelds whose residue ﬁelds are algebraic over F p , i.e., a class of ﬁelds for which a relative ver- sion of the Grothendieck Conjecture for hyperbolic curves has been known for some time [cf. [AnabTop], Theorem 4.12]. Theorem 3.13 (Absolute version of the Grothendieck Conjecture for conﬁguration spaces associated to arbitrary hyperbolic curves over p-adic local ﬁelds). Let p † , p ‡ be prime numbers; K † , K ‡ mixed characteristic local ﬁelds of residue characteristic p † , p ‡ , respectively; X † , X ‡ hyperbolic curves over K † , K ‡ , respectively; n † , n ‡ positive integers. Write X n † † (respectively, X n ‡ ‡ ) for the n † -th (respectively, n ‡ -th) conﬁguration space associated to X † (respectively, X ‡ ). Then the natural map Isom(X n † † , X n ‡ ‡ ) −→ OutIsom(Π X † , Π X ‡ ) n † n ‡ is bijective. Proof. First, we observe that any isomorphism of schemes between X n † † and X n ‡ ‡ necessarily lies over an isomorphism of ﬁelds between K † and K ‡ . [Indeed, this follows immediately by a similar argument to the argument applied in the proof of Theorem 3.12.] Now Theorem 3.13 follows immediately from a def routine argument via induction on n = n † = n ‡ [cf. [AbsTopI], Theorem 2.6, (v); [HMM], Theorem A, (i), (ii)], by combining Theorem 3.12 of the present paper with the relative version of the Grothendieck Conjecture given in [LocAn], Theorem A. Next, we discuss the functorial behavior of the lengths of nodes of special ﬁbers of compactiﬁed semistable models [cf. Deﬁnition 3.14 below] with respect to ﬁnite morphisms between compactiﬁed semistable models that extend ﬁnite étale Galois coverings of hyperbolic curves over mixed characteristic complete discrete valuation ﬁelds. Deﬁnition 3.14. Let K be a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; X a hyperbolic curve over K; X a compactiﬁed semistable model with split reduction of X over O K ; e a node of X s . Recall that the completion of the local ring O X ,e at e is isomorphic to O K [[x, y]]/(xy − a), where x, y denote indeterminates; a ∈ m K \ {0}. Then we shall refer to v p (a) as the length of e. [Note that the length of e is independent of the choice of a, ∼ as well as of the isomorphism O X ,e → O K [[x, y]]/(xy − a) over O K [cf. [Hur], §3.7].] 92 Proposition 3.15 (Functorial behavior of the lengths of nodes). Let K be a mixed characteristic complete discrete valuation ﬁeld of residue character- istic p; X a hyperbolic curve over K; Y → X a [connected] ﬁnite étale Galois covering of hyperbolic curves over K; Y a compactiﬁed semistable model of Y def over O K that is stabilized by G = Gal(Y /X). Write X for the compactiﬁed semistable model of X over O K obtained by forming the quotient of Y by the action of G on Y [cf. Proposition 2.3, (iv)]; f : Y → X for the natural quotient morphism. Suppose that • Y has split reduction, and that • the natural action of G on Y s does not permute the branches of some node e Y of Y s . def Write e X = f (e Y ) for the node of X s determined by e Y [cf. Proposition 2.3, (iv)]; l X , l Y for the lengths of the nodes e X , e Y [relative to the compactiﬁed semistable models X , Y, respectively]. Then there exists a positive integer m such that l X = m · l Y . Moreover, the positive integer m may be computed as the cardinality of the decomposition subgroup [i.e., the stabilizer subgroup] of e Y in G. def Proof. Write S = Spec O K ; S log for the log scheme obtained by equipping S with the log structure determined by the closed point of S; X log , Y log for the log schemes over S log determined by the compactiﬁed semistable models X , Y, respectively [cf. the discussion of the subsection in Notations and Conventions entitled “Log schemes”]. Observe that f naturally determines a ﬁnite morphism f log : Y log → X log of log schemes, hence a ﬁnite morphism X ,e ) log , Y,e ) log −→ (Spec O (Spec O Y X Y,e denote the completions of the respective normal local rings X ,e , O where O X Y O X ,e X , O Y,e Y , and the superscripts “log” denote the log structures induced by the respective log structures of X log , Y log . Since Y is assumed to have split reduction, it follows [cf. the discussion of Deﬁnition 3.14] that X ,e ∼ O K [[u 1 , u 2 ]]/(u 1 u 2 − a), O X = Y,e ∼ O K [[v 1 , v 2 ]]/(v 1 v 2 − b), O Y = where u 1 , u 2 , v 1 , v 2 denote indeterminates; a, b ∈ m K \{0} are elements such that l X = v p (a), l Y = v p (b). Moreover, it follows immediately from the deﬁnitions of the log structures involved, together with the geometry of the irreducible X ,e and Spec O Y,e , that, after components of the special ﬁbers of Spec O X Y possibly switching the indices ∈ {1, 2} of [either or both of] the pairs (u 1 , u 2 ) × and (v 1 , v 2 ), there exist positive integers m 1 , m 2 and units c 1 , c 2 ∈ O Y,e Y such that m 1 ≥ m 2 , and u 1 = c 1 · v 1 m 1 , u 2 = c 2 · v 2 m 2 , 93 Y,e via the natural injection O X ,e → where we regard u 1 , u 2 as elements in O Y X O Y,e Y . In particular, it holds that a = u 1 u 2 = c 1 c 2 v 1 m 1 v 2 m 2 = c 1 c 2 v 1 m 1 −m 2 b m 2 ∈ O K [[v 1 , v 2 ]]/(v 1 v 2 − b). On the other hand, such a relation implies, in light of the well-known structure of the log structures involved, i.e., in eﬀect, the geometry of the irreducible X ,e and Spec O Y,e , that m def components of the special ﬁbers of Spec O = X Y m m 1 = m 2 , hence that a = c 1 c 2 b . In particular, we conclude that l X = m · l Y , as desired. Finally, the fact that m may be computed as the cardinality of the decomposition subgroup [i.e., the stabilizer subgroup] of e Y in G follows immediately from the fact that the generic degree of the [ﬁnite, generically Y,e → Spec O X ,e is [easily computed, via the explicit étale] morphism Spec O Y X presentations of O Y,e Y , O X ,e X given above, to be] m. This completes the proof of Proposition 3.15. Next, we apply Proposition 3.15, together with the theory of p-adic arith- metic cuspidalizations developed in [Tsjm], §2, to prove that the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature [cf. [Tsjm], Remark 2.1.2] in fact coincide. Theorem 3.16 (Equality of various p-adic versions of the Grothendieck- def -Teichmüller group). Write X = P 1 Q \ {0, 1, ∞}; p GT ⊆ Out(Π X ) for the Grothendieck-Teichmüller group [cf. [CmbCsp], Remark 1.11.1]; GT M ⊆ GT (⊆ Out(Π X )) for the metrized Grothendieck-Teichmüller group [cf. [CbTpIII], Remark 3.19.2]; def GT tp = GT ∩ Out(Π tp p X ) ⊆ Out(Π X ) [cf. the subsection in Notations and Conventions entitled “Fundamental groups”; [Tsjm], Deﬁnition 2.1]. Then the natural inclusion GT M ⊆ GT tp p of subgroups of GT is an equality. In particular, it holds that GT M = GT p = GT G = GT tp p [cf. [Tsjm], Remark 2.1.2]. 94 Proof. First, we recall that there exists a natural surjection φ : GT tp p G Q p whose restriction to G Q p is the identity automorphism [cf. [Tsjm], Corollary B, as well as Remark 3.16.1 below]. Thus, since G Q p ⊆ GT M ⊆ GT tp p , it suﬃces to prove that Ker(φ) ⊆ GT M . Let σ ∈ Ker(φ). Fix a lifting σ̃ ∈ Aut(Π tp X ) of will always denote the Primes- σ. [Here and in the following discussion, Π tp (−) tempered fundamental group of (−).] Then it follows immediately from the construction of φ [cf. the discussion, in the proof of [Tsjm], Corollary 2.4, of the two paragraphs following the proof of Claim 2.4.B; the discussion, in the proof of [HMT], Theorem 4.4, of the observa- tion immediately following the statement of Claim 4.4.A; [NCBel], Corollary 1.2] that, for any ﬁnite subset S ⊆ Q \ {0, 1} ⊆ X(Q p ), σ̃ lifts to an automorphism def of Π tp X S [where we write X S = X \ S] with respect to the natural surjection tp Π tp X S Π X determined [up to composition with an inner automorphism] by the natural open immersion X S → X. Next, let ψ Y : Y → X be a connected ﬁnite étale covering over Q p . Write ψ Z : Z → X for the Galois closure of ψ Y ; Z for the compactiﬁed stable model of σ Z over O Q p ; ψ Y σ : Y σ → X, ψ Z : Z σ → X for the connected ﬁnite étale coverings tp tp tp over Q p that correspond to the open subgroups σ̃(Π tp Y ) ⊆ Π X , σ̃(Π Z ) ⊆ Π X , respectively. For each ﬁnite subset S ⊆ Q \ {0, 1} ⊆ X(Q p ), write def • Z S (respectively, Z S σ ) for the compactiﬁed stable model of Z S = Z \ −1 σ −1 (S) (respectively, Z S σ = Z σ \ (ψ Z ) (S)) over O Q p ; ψ Z def • X S (respectively, Y S , Y S σ ) for the compactiﬁed semistable model of X S (respectively, Y S = Y \ ψ Y −1 (S), Y S σ = Y σ \ (ψ Y σ ) −1 (S)) obtained by forming the quotient of Z S (respectively, Z S , Z S σ ) via the natural action of Gal(Z/X) (respectively, Gal(Z/Y ), Gal(Z σ /Y σ )) [cf. Proposition 2.3, (iv)]. def def Next, observe that there exists a ﬁnite subset T ⊆ Q \ {0, 1} ⊆ X(Q p ) such that • the natural action of Gal(Z/X) on (Z T ) s does not permute any branches of nodes, and • X T , Y T , Y T σ are the respective compactiﬁed stable models of X T , Y T , Y T σ over O Q p . Then since σ̃ lifts to an automorphism of Π tp X T [cf. the above discussion], hence to an isomorphism ∼ tp tp tp tp tp Π tp Z × Π tp Π X T = Π Z T → Π Z σ = Π Z σ × Π tp Π X T , T X 95 X it follows immediately from Proposition 2.3, (iv), together with [SemiAn], Corol- lary 3.11, that σ̃ induces a commutative diagram of semi-graphs ∼ Γ Z T −−−−→ Γ Z T σ ⏐ ⏐ ⏐ ⏐ Γ X T Γ X T , where Γ (−) denotes the dual semi-graph associated to (−) s , compatible with the respective natural actions of Gal(Z/X), Gal(Z σ /X). Thus, we conclude ∼ from Proposition 3.15 that the isomorphism Γ Z T → Γ Z T σ of dual semi-graphs is compatible with the respective metric structures [cf. [CbTpIII], Deﬁnition 3.5, (iii)]. On the other hand, σ̃ also induces a commutative diagram of semi-graphs ∼ Γ Z T −−−−→ Γ Z T σ ⏐ ⏐ ⏐ ⏐ ∼ Γ Y T −−−−→ Γ Y T σ compatible with the respective natural actions of Gal(Z/Y ), Gal(Z σ /Y σ ) [cf. Proposition 2.3, (iv); [SemiAn], Corollary 3.11]. Thus, since the isomorphism ∼ Γ Z T → Γ Z T σ of dual semi-graphs is compatible with the respective metric struc- ∼ tures, we conclude from Proposition 3.15 again that the isomorphism Γ Y T → Γ Y T σ of dual semi-graphs is also compatible with the respective metric struc- tures. Finally, it follows immediately from the well-known theory of pointed stable curves and contraction morphisms that arise from eliminating cusps, as exposed in [Knud] [cf. also Remark 2.1.4], that this implies that, if we write Y, Y σ for the respective compactiﬁed stable models of Y , Y σ over O Q p , then ∼ the isomorphism Γ Y → Γ Y σ of dual semi-graphs induced by σ̃ [cf. [SemiAn], Corollary 3.11] is compatible with the respective metric structures. Thus, we conclude from [CbTpIII], Deﬁnition 3.7, (ii); [CbTpIII], Remark 3.19.2, that GT M = GT tp p . This completes the proof of Theorem 3.16. Remark 3.16.1. Here, we recall that one of the key ingredients in the proof of [Tsjm], Corollary B, is the theory of resolution of nonsingularities developed in [Lpg1]. As a corollary, we obtain the following aﬃrmative answer to the question posed in the discussion immediately preceding Theorem E in [CbTpIII], Intro- duction: Corollary 3.17 (Commensurable terminality of various p-adic versions of the Grothendieck-Teichmüller group). We maintain the notation of Theorem 3.16. Then GT M = GT p = GT G = GT tp p is commensurably terminal M in GT, i.e., the commensurator C GT (GT ) of GT M in GT is equal to GT M . 96 Proof. It follows immediately from Theorem 3.16, together with [CbTpIII], The- orem E, that GT M ⊆ C GT (GT M ) ⊆ GT G = GT M . Thus, we conclude that C GT (GT M ) = GT M , as desired. Proposition 3.18 (Reconstruction of the subset of Q p -rational points def from the p-adic Grothendieck-Teichmüller group). Write X = P 1 Q \ p {0, 1, ∞}; Π tp X for the Primes-tempered fundamental group of X. Then the subset X(Q p ) ⊆ X(C p ), where we think of “X(C p )” as the set reconstructed from Π tp X in Corollary 3.10, may be reconstructed, in a purely combinatorial/group- theoretic way, from the data tp tp (Π tp X , GT p ⊆ Out(Π X )) tp — consisting of the underlying topological group of Π tp X and the subgroup GT p ⊆ tp Out(Π X ) — as the subset of elements ﬁxed by some open subgroup of GT tp p . Moreover, this reconstruction procedure is functorial with respect to isomor- phisms of topological groups for which the induced isomorphism on “Out(−)” preserves the given subgroup of “Out(−)”. Proof. First, we observe that it follows immediately from the existence of the natural homeomorphism “θ X ” of Proposition 2.3, (viii), together with the deﬁ- tor ” [cf. Deﬁnition 2.2, (vi)], that the subset X(Q p ) ⊆ X(C p ) nition of “VE( X) is dense in X(C p ), and that the natural action of GT tp p on X(C p ) is via self- homeomorphisms of X(C p ) [cf. Corollary 3.10 and its proof; Corollary 3.16]. Thus, since the natural action of GT tp p on X(Q p ) factors through the surjection tp GT p G Q p [cf. [Tsjm], Corollary B, and its proof], we conclude that the tp natural action GT tp p on X(C p ) factors through this surjection GT p G Q p , and hence [cf. [Tate], §3.3, Theorem 1] that the subset X(Q p ) ⊆ X(C p ) may be characterized as the subset of elements ﬁxed by some open subgroup of GT tp p . This completes the proof of Proposition 3.18. Finally, we apply the theory of resolution of nonsingularities and point- theoreticity [cf., especially, Corollary 2.5, (i); Corollary 3.10], together with the theory of metric-admissibility developed in [CbTpIII], §3, to construct certain arithmetic cuspidalizations of the [Primes-]tempered fundamental groups of hy- perbolic curves over Q p equipped with “proj-metric structures” [cf. Deﬁnition 3.19 below]. Deﬁnition 3.19. Let Σ ⊆ Primes be a subset of cardinality ≥ 2 that con- tains p; K a mixed characteristic complete discrete valuation ﬁeld of residue characteristic p; X a hyperbolic curve over K. Write Π tp X for the Σ-tempered ∗ fundamental group of X. For each open subgroup Π ⊆ Π tp X of ﬁnite index, write 97 • X Π ∗ for the compactiﬁed stable model over O K of the hyperbolic curve over K corresponding to the open subgroup Π ∗ ⊆ Π tp X ; • Γ Π ∗ for the dual graph associated to (X Π ∗ ) s ; • μ Π ∗ for the metric structure on Γ Π ∗ associated to X Π ∗ , considered up to multiplication by a constant ∈ Q × [cf. [CbTpIII], Deﬁnition 3.5, (iii)]. Then we shall refer to μ Π ∗ as the proj-metric structure on Γ Π ∗ . We shall refer to the collection of data of proj-metric structures {μ Π ∗ } associated to the charac- teristic open subgroups {Π ∗ ⊆ Π tp X } of ﬁnite index as the proj-metric structure tp on Π X . Theorem 3.20 (Construction of certain arithmetic cuspidalizations of geometric tempered fundamental groups). Let K be a mixed characteris- tic complete discrete valuation ﬁeld of residue characteristic p; X a hyperbolic a universal pro-Primes covering of X. Suppose that X sat- curve over K; X isﬁes Primes-RNS. Write Ω for the p-adic completion of K. For n ≥ 2 an def integer, write X n for the n-th conﬁguration space associated to X; Π 1 = Π X ; def Π n = Π X n ; Π 2/1 for the kernel of the natural surjection Π 2 Π 1 induced by the ﬁrst projection X 2 → X, where we regard Π 2 as a quotient of Π n via the projection X n → X 2 to the ﬁrst two factors; Π tp 1 for the Primes-tempered fundamental group of X; def (Out(Π n ) ⊇) Out(Π n ) tp = Out gF (Π n ) × Out(Π 1 ) Out(Π tp 1 ), def (Out(Π n ) ⊇) Out gFC (Π n ) = Out gF (Π n ) ∩ Out FC (Π n ) (⊆ Out(Π 1 )), def (Out(Π n ) ⊇) Out gFC (Π n ) M = Out gF (Π n ) ∩ Out FC (Π n ) M (⊆ Out(Π tp 1 ) ⊆ Out(Π 1 )) [cf. [HMM], Deﬁnition 2.1, (iv); [CbTpI], Theorem A, (i); [CbTpIII], Propo- sition 3.3, (iv); [CbTpIII], Deﬁnition 3.7, (i), (ii), (iii); [NodNon], Theorem B]; VE(Π tp Π tp 1 ), 1 (Ω) and “ X(Ω)” for the respective sets “VE( X)” equipped with their natural actions tp by Aut(Π 1 ) and Π 1 constructed in [the proof of ] Corollary 2.5, (i), and Corol- tp lary 3.10 from the underlying topological group of Π tp 1 . Let x̃ ∈ VE(Π 1 ). Then the following hold: (i) One may construct an “arithmetic cuspidalization” of Π tp 1 associated to x̃ from the data consisting of • the topological group Π n equipped with the quotients Π n Π 2 Π 1 and a topology [i.e., the tempered topology] on the subquotient Π n Π 2 Π 1 ⊇ Π tp 1 98 in a fashion that is functorial with respect to isomorphisms of this data [in the evident sense] as follows: Observe that the subgroup Out(Π n ) tp ⊆ Out(Π n ) may be constructed from the given data [cf. [HMM], Theorem A, (ii)]. Write tp tp out tp n D x̃ ⊆ Π 1 Out(Π n ) tp = Aut(Π tp 1 ) × Out(Π tp ) Out(Π n ) 1 [cf. [CbTpIII], Proposition 3.3, (i), (ii); [MT], Proposition 2.2, (ii)] for the stabilizer subgroup of x̃. Note that there exists a natural exact sequence [that may be constructed from the given data] out out tp tp 1 −→ Π 2/1 −→ (Π 2 × Π 1 Π tp −→ Π tp −→ 1. 1 ) Out(Π n ) 1 Out(Π n ) Thus, by pulling-back the above exact sequence via the inclusion n D x̃ tp ⊆ out tp Π tp 1 Out(Π n ) , we obtain an exact sequence out 1 −→ Π 2/1 −→ Π 2/1 n D x̃ tp −→ n D x̃ tp −→ 1. out We shall refer to Π 2/1 n D x̃ tp as the [n-th] arithmetic cuspidalization of Π tp 1 associated to x. (ii) Write n-alg Π tp ⊆ Π tp 1 (Ω) 1 (Ω) for the subset of elements ξ ∈ Π tp 1 (Ω) whose Π 1 -orbit Π 1 · ξ is stabilized by some open subgroup of Out gFC (Π n ) M (⊆ Out(Π tp 1 )) [cf. Remark 3.20.1 n-alg ⊆ Π tp below]. Suppose that x̃ arises from an element ∈ Π tp 1 (Ω) 1 (Ω), which, by a slight abuse of notation, we shall also denote by x̃. Write def X x = X Ω \ {x}, where x ∈ X(Ω) denotes the element determined by x̃. Then the [Primes-]tempered fundamental group Π tp X x (⊆ Π 2/1 ) of X x [where we identify Π 2/1 with Π X x ], together with the proj-metric structure on Π tp X x , may be reconstructed, in a purely combinatorial/group- theoretic way, from the following data • the topological group Π n equipped with the quotients Π n Π 2 Π 1 and a topology [i.e., the tempered topology] and proj-metric structure on the subquotient Π n Π 2 Π 1 ⊇ Π tp 1 ; • the subgroup Out gFC (Π n ) ⊆ Out(Π n ) [cf. Remark 3.20.2 below] in a fashion that is functorial with respect to isomorphisms of this data [in the evident sense]. 99 Proof. Assertion (i) follows immediately from the various deﬁnitions involved. Next, we verify assertion (ii). Write M def = n D x̃ out gFC Π tp (Π n ) M 1 Out ∩ n D x̃ tp out tp (⊆ Π tp 1 Out(Π n ) ). In particular, it follows immediately from assertion (i), together with the various deﬁnitions involved, that one may construct, from the given data, n D x̃ M , together with the natural outer action of n D x̃ M on Π 2/1 . Let Π † 2/1 ⊆ Π 2/1 be an open subgroup that is normal in Π 2 ; l a prime number = p. Write K ⊆ K tm (⊆ K) def tm /K ur ); Π 2/1 Π ∗ 2/1 for for the maximal tame extension of K; G tm K = Gal(K the maximal almost pro-l quotient associated to [i.e., “with respect to”] Π † 2/1 ⊆ Π 2/1 [cf. [CbTpIII], Deﬁnition 1.1]. Let us assume further that the quotient Π 2 Π 2 /Ker(Π 2/1 Π ∗ 2/1 ) is F-characteristic [cf. [CbTpIII], Deﬁnition 2.1, (iii)]. Note that this implies that, relative to the identiﬁcation of Π 2/1 with Π X x [cf. the statement of assertion (ii)], the natural Π 2/1 -outer action of G K on Π 2 [which is well-deﬁned after possibly replacing K by a suitable ﬁnite extension ∗ ﬁeld of K] induces a natural outer action of G K on Π 2/1 . Thus, in order to complete the proof of assertion (ii), it suﬃces, in light of the argument given in the proof of [CbTpIII], Theorem 3.9 [cf., especially, the equivalence stated in the ﬁnal display of the proof of [CbTpIII], Theorem 3.9], to reconstruct, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, the image of G tm K in Out(Π ∗ 2/1 ) via the natural outer representation. Next, we verify the following assertion: →) Y → X be a [connected] ﬁnite étale Galois Claim 3.20.A: Let ( X covering over K. Then there exist a compactiﬁed semistable model → Y of Y over O K and a [connected] ﬁnite étale Galois covering ( X ) Z → X over K that dominates Y → X and satisﬁes the following conditions: • The compactiﬁed stable model Z of Z over O K dominates Y. • The component of the VE-chain x̃ corresponding to Z is an irreducible component of Z s that maps to a smooth closed point ∈ Y s that is not a cusp. Indeed, since X satisﬁes Primes-RNS, Claim 3.20.A follows immediately from the fact that the VE-chain x̃ arises from an element ∈ Π tp 1 (Ω) that is of type 1, hence weakly verticial [cf. Proposition 2.4, (v); Remark 3.1.1; Proposition 3.3, (ii); Proposition 3.4, (iii); the theory of pointed stable curves, as exposed in [Knud]]. Next, we observe that it follows from [CbTpIII], Proposition 2.3, (ii) [cf. conditions (a), (b), (c) below]; [CbTpIII], Corollary 2.10 [cf. condition (c) be- low], together with Claim 3.20.A [cf. condition (b) below], that, after possibly replacing Π † 2/1 ⊆ Π 2/1 by a smaller open subgroup that satisﬁes the same con- ditions as Π † 2/1 , there exist F-characteristic SA-maximal almost pro-l quotients Π 2 Π ∗ 2 , Π 2 Π ∗∗ 2 satisfying the following conditions: 100 (a) The F-characteristic SA-maximal almost pro-l quotient Π 2 Π ∗∗ 2 dom- inates the F-characteristic SA-maximal almost pro-l quotient Π 2 Π ∗ 2 . In particular, we obtain a commutative diagram of proﬁnite groups ∗∗ ∗∗ 1 −−−−→ Π ∗∗ 2/1 −−−−→ Π 2 −−−−→ Π 1 −−−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1 −−−−→ Π ∗ 2/1 −−−−→ Π ∗ 2 −−−−→ Π ∗ 1 −−−−→ 1, ∗ ∗∗ where the quotients Π ∗ 1 , Π ∗∗ 1 of Π 1 induced by Π 2 , Π 2 are the center-free [cf. [CbTpIII], Proposition 1.7, (i)] maximal almost pro-l quotients of Π 1 associated to normal open subgroups of Π 1 ; the quotients Π ∗ 2/1 , Π ∗∗ 2/1 of Π 2/1 induced by Π ∗ 2 and Π ∗∗ are the center-free [cf. [CbTpIII], Proposition 2 1.7, (i)] maximal almost pro-l quotients of Π 2/1 associated to normal open subgroups of Π 2/1 ; the vertical arrows denote surjective homomorphisms. (b) Fix a normal open subgroup Π Y ⊆ Π 1 whose associated maximal almost pro-l quotient coincides with Π 1 Π ∗ 1 . Then there exists a normal open subgroup Π Z ⊆ Π 1 such that: • It holds that Π Z ⊆ Π Y . [In particular, the maximal almost pro-l quotient associated to the normal open subgroup Π Z ⊆ Π 1 dominates the maximal almost pro-l quotient Π 1 Π ∗ 1 .] • The maximal almost pro-l quotient Π 1 Π ∗∗ 1 dominates the maximal almost pro-l quotient associated to the normal open subgroup Π Z ⊆ Π 1 . →) Y → X, ( X →) Z → X for the respective [connected] • Write ( X ﬁnite étale Galois coverings over K associated to the normal open subgroups Π Y ⊆ Π 1 , Π Z ⊆ Π 1 . Then there exists a compactiﬁed semistable model Y of Y over O K such that the compactiﬁed stable model Z of Z over O K dominates Y, and, moreover, the component of the VE-chain x̃ corresponding to Z is an irreducible component of Z s that maps to a smooth closed point ∈ Y s that is not a cusp. FC ∗ ∗∗ (Π ∗∗ (c) Every element ∈ Out FC (Π ∗∗ 2 Π 2 ) ∩ Ker(Out 2 ) → Out(Π 1 )) [cf. [CbTpIII], Deﬁnition 2.1, (viii)] induces the trivial outer automorphism of Π ∗ 2 . Write D x̃ ∗∗ out out for the image of n D x̃ M (⊆ Π 1 Out gFC (Π n ) ⊆ Π 1 Out gFC (Π 2 )) [where the second inclusion follows from [NodNon], Theorem B] via the natural homomor- out out FC phism Π 1 Out gFC (Π 2 ) → Π ∗∗ (Π ∗∗ 1 Out 2 ); out FC ∗ ∗∗ ρ ∗∗ : D x̃ ∗∗ ⊆ Π ∗∗ (Π ∗∗ 1 Out 2 Π 2 ) −→ Out(Π 1 ) 101 for the natural composite homomorphism. Next, we verify the following assertion: Claim 3.20.B: There exists an open subgroup D ∗∗ ⊆ D x̃ ∗∗ such that every element ∈ D ∗∗ ∩ Ker(ρ ∗∗ ) induces the trivial outer action on Π ∗ 2/1 . Indeed, observe that since Π ∗∗ 1 is center-free [cf. condition (a)], the natural homomorphism out ∗∗ ∗∗ D x̃ ∗∗ −→ Π ∗∗ 1 Out(Π 1 ) = Aut(Π 1 ) induces a natural homomorphism φ : Ker(ρ ∗∗ ) −→ Π ∗∗ 1 . Thus, it follows immediately from the ﬁnal portion of condition (b) [cf. also the proof of Claim 3.20.A; our assumption that X satisﬁes Primes-RNS; the portion of Proposition 3.5, (i), concerning the weakly verticial case], together with the various deﬁnitions involved [cf., especially, the deﬁnition of D x̃ ∗∗ ], that the image of the natural composite homomorphism φ ∗ Ker(ρ ∗∗ ) −→ Π ∗∗ 1 Π 1 is ﬁnite. In particular, there exists an open subgroup D ∗∗ ⊆ D x̃ ∗∗ such that every element ∈ D ∗∗ ∩ Ker(ρ ∗∗ ) induces the trivial automorphism of Π ∗ 1 . On the other hand, this implies, in light of condition (c), that every element ∈ D ∗∗ ∩ Ker(ρ ∗∗ ) induces the trivial outer automorphism of Π ∗ 2 . Thus, since Π ∗ 1 is center-free [cf. condition (a)], we conclude that every element ∈ D ∗∗ ∩ Ker(ρ ∗∗ ) induces the trivial outer automorphism of Π ∗ 2/1 . This completes the proof of Claim 3.20.B. Next, let us observe that, by applying the argument given in the proof of Theorem 3.9 [cf., especially, the equivalence stated in the ﬁnal display of the proof of [CbTpIII], Theorem 3.9], together with condition (a), we conclude that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, one tm may reconstruct, from the proj-metric structure on Π tp 1 , the image I of G K in Out(Π ∗∗ ) via the natural outer representation. On the other hand, observe 1 that ρ ∗∗ (D ∗∗ ) contains an open subgroup of I [cf. Remark 3.20.1 below]. Thus, since D ∗∗ ∩ Ker(ρ ∗∗ ) induces the trivial outer action on Π ∗ 2/1 [cf. Claim 3.20.B], by considering the outer action of D ∗∗ ∩ (ρ ∗∗ ) −1 (I) on Π ∗ 2/1 , we conclude that, after possibly replacing K by a suitable ﬁnite extension ﬁeld of K, one may ∗ reconstruct the image of G tm K in Out(Π 2/1 ), as desired. This completes the proof of assertion (ii), hence of Theorem 3.20. 102 Remark 3.20.1. Suppose that we are in the situation of Theorem 3.20, (ii). Then it follows immediately from the deﬁnitions [cf. also the natural homomorphism G K → Out gFC (Π n ) M [which is well-deﬁned after possibly replacing K by a suit- able ﬁnite extension ﬁeld of K]; the proof of Theorem 3.11, (ii)] that the subset tp tp n-alg Π tp ⊆ Π tp 1 (Ω) 1 (Ω) is contained in the subset Π 1 (K) ⊆ Π 1 (Ω) correspond- constructed in Corollary 3.10. ing to the K-rational points of the set “ X(Ω)” Moreover, it follows immediately from Proposition 3.18 [cf. also Theorem 3.16; [HMM], Corollaries B, C] that the inclusion n-alg ⊆ Π tp Π tp 1 (Ω) 1 (K) is in fact an equality in the case where X = P 1 Q \ {0, 1, ∞}. It is not clear to p the authors, however, at the time of writing of the present paper whether or not the inclusion of the above display is an equality in general. Remark 3.20.2. Suppose that we are in the situation of Theorem 3.20, (ii). Then we observe that the data “Out gFC (Π n ) ⊆ Out(Π n )” may be omitted from the list of data in the statement of Theorem 3.20, (ii), in either of the following situations [cf. [CbTpII], Theorem A, (ii); [HMM], Theorem A, (ii); [HMT], Corollary 2.2]: • X is of genus 0. • X is aﬃne, and n ≥ 3. • X is proper, and n ≥ 4. It is not clear to the authors, however, at the time of writing of the present paper whether or not this data “Out gFC (Π n ) ⊆ Out(Π n )” may be omitted in general. References [André] Y. André, On a geometric description of Gal(Q p /Q p ) and a p-adic , Duke Math. J. 119 (2003), pp. 1–39. avatar of GT [Brk] V. G. Berkovich, Spectral theory and analytic geometry over non- Archimedean ﬁelds, Mathematical Surveys and Monographs, 33, American Mathematical Society (1990). [BLR] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer-Verlag (1990). [DM] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, IHES Publ. Math. 36 (1969), pp. 75–109. 103 [EP] A. Engler and A. Prestel, Valued Fields, Springer Monographs in Mathe- matics, Springer-Verlag (2005). [FC] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete 22, Springer-Verlag (1990). [EGAIV 2 ] A. Grothendieck, Éléments de géometrie algébrique. IV: Étude locale des schémas et des morphismes de schémas, IHES Publ. Math. 24 (1965). [EGAIV 3 ] A. Grothendieck, Éléments de géometrie algébrique. IV: Étude locale des schémas et des morphismes de schémas, IHES Publ. Math. 28 (1966). [Hgsh] K. Higashiyama, The semi-absolute anabelian geometry of geometrically pro-p arithmetic fundamental groups of associated low-dimensional conﬁg- uration spaces, Publ. Res. Inst. Math. Sci. 58 (2022), pp. 579–634. [BSC] Y. Hoshi, Conditional results on the birational section conjecture over small number ﬁelds, Automorphic forms and Galois representations, Vol. 2, London Math. Soc. Lecture Note Ser. 415, Cambridge Univ. Press (2014), pp. 187–230. [NodNon] Y. Hoshi and S. Mochizuki, On the combinatorial anabelian geometry of nodally nondegenerate outer representations, Hiroshima Math. J. 41 (2011), pp. 275–342. [CbTpI] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial an- abelian geometry of hyperbolic curves I: Inertia groups and proﬁnite Dehn twists, Galois-Teichmüller Theory and Arithmetic Geometry, Adv. Stud. Pure Math. 63, Math. Soc. Japan (2012), pp. 659–811. [CbTpII] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial an- abelian geometry of hyperbolic curves II: Tripods and combinatorial cusp- idalization, Lecture Notes in Mathematics 2299, Springer-Verlag (2022). [CbTpIII] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves III: Tripods and tempered funda- mental groups, RIMS Preprint 1763 (November 2012). [CbTpIV] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves IV: Discreteness and sections, to appear in Nagoya Math. J. [Knud] F. F. Knudsen, The projectivity of the moduli space of stable curves, II: The stacks M g,n , Math. Scand. 52 (1983), pp. 161–199. [HMM] Y. Hoshi, A. Minamide, and S. Mochizuki, Group-theoreticity of numer- ical invariants and distinguished subgroups of conﬁguration space groups, Kodai Math. J. 45 (2022), pp. 295–348. 104 [HMT] Y. Hoshi, S. Mochizuki, and S. Tsujimura, Combinatorial construction of the absolute Galois group of the ﬁeld of rational numbers, RIMS Preprint 1935 (December 2020). [Lpg1] E. Lepage, Resolution of non-singularities for Mumford curves, Publ. Res. Inst. Math. Sci. 49 (2013), pp. 861–891. [Lpg2] E. Lepage, Resolution of non-singularities and the absolute anabelian conjecture, arXiv:2306.07058 [math.AG]. [Milne] J. S. Milne, Jacobian varieties in Arithmetic Geometry, ed. by G. Cornell and J.H. Silverman, Springer-Verlag (1986), pp. 167–212. [MiSaTs] A. Minamide, K. Sawada, and S. Tsujimura, On generalizations of anabelian group-theoretic properties, RIMS Preprint 1965 (August 2022). [MiTs1] A. Minamide and S. Tsujimura, Anabelian group-theoretic properties of the absolute Galois groups of discrete valuation ﬁelds, J. Number Theory 239 (2022), pp. 298–334. [MiTs2] A. Minamide and S. Tsujimura, Anabelian geometry for Henselian dis- crete valuation ﬁelds with quasi-ﬁnite residues, RIMS Preprint 1973 (June 2023). [Hur] S. Mochizuki, The geometry of the compactiﬁcation of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), pp. 355–441. [PrfGC] S. Mochizuki, The proﬁnite Grothendieck conjecture for closed hyper- bolic curves over number ﬁelds, J. Math. Sci. Univ. Tokyo 3 (1996), pp. 571–627. [LocAn] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), pp. 319–423. [ExtFam] S. Mochizuki, Extending families of curves over log regular schemes, J. reine angew. Math. 511 (1999), pp. 43–71. [CanLift] S. Mochizuki, The absolute anabelian geometry of canonical curves, Kazuya Kato’s ﬁftieth birthday, Doc. Math. 2003, Extra Vol., pp. 609–640. [AnabTop] S. Mochizuki, Topics surrounding the anabelian geometry of hyper- bolic curves, Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ. 41, Cambridge Univ. Press. (2003), pp. 119–165. [NCBel] S. Mochizuki, Noncritical Belyi maps, Math. J. Okayama Univ. 46 (2004), pp. 105–113. [AbsAnab] S. Mochizuki, The absolute anabelian geometry of hyperbolic curves, Galois Theory and Modular Forms, Kluwer Academic Publishers (2004), pp. 77–122. 105 [SemiAn] S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), pp. 221–322. [CmbGC] S. Mochizuki, A combinatorial version of the Grothendieck conjec- ture, Tohoku Math. J. 59 (2007), pp. 455–479. [CmbCsp] S. Mochizuki, On the combinatorial cuspidalization of hyperbolic curves, Osaka J. Math. 47 (2010), pp. 651–715. [AbsTopI] S. Mochizuki, Topics in absolute anabelian geometry I: Generalities, J. Math. Sci. Univ. Tokyo 19 (2012), pp. 139–242. [AbsTopII] S. Mochizuki, Topics in absolute anabelian geometry II: Decompo- sition groups and endomorphisms, J. Math. Sci. Univ. Tokyo 20 (2013), pp. 171–269. [AbsTopIII] S. Mochizuki, Topics in absolute anabelian geometry III: Global reconstruction algorithms, J. Math. Sci. Univ. Tokyo 22 (2015), pp. 939– 1156. [MT] S. Mochizuki and A. Tamagawa, The algebraic and anabelian geometry of conﬁguration spaces, Hokkaido Math. J. 37 (2008), pp. 75–131. [Neu] J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Springer-Verlag (1999). [NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number ﬁelds, Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag (2000). [Ray1] M. Raynaud, Sections des ﬁbrés vectoriels sur une courbe, Bull. Soc. Math. France, 110 (1982), pp. 103–125. [Ray2] M. Raynaud, p-groupes et réduction semi-stable des courbes, The Grothendieck Festschrift, Vol. III, Progr. Math., 88, Birkhäuser (1990), pp. 179–197. [Tama1] A. Tamagawa, The Grothendieck conjecture for aﬃne curves, Compo- sitio Math. 109 (1997), pp. 135–194. [Tama2] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ. Res. Inst. Math. Sci. 40 (2004), pp. 1291–1336. [Tate] J. Tate, p-divisible groups, Proceedings of a Conference on Local Fields, Driebergen, Springer-Verlag (1967), pp. 158–183. [Tsjm] S. Tsujimura, Combinatorial Belyi cuspidalization and arithmetic sub- quotients of the Grothendieck-Teichmüller group, Publ. Res. Inst. Math. Sci. 56 (2020), pp. 779–829. 106 Updated versions of [HMT], [CbTpIII], [CbTpIV] may be found at the following URL: http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html (Shinichi Mochizuki) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Email address: motizuki@kurims.kyoto-u.ac.jp (Shota Tsujimura) Research Institute for Mathematical Sciences, Kyoto Uni- versity, Kyoto 606-8502, Japan Email address: stsuji@kurims.kyoto-u.ac.jp 107